Find the Inverse 5x^(1/5)+3

$5 x^{\frac{1}{5}} + 3$

Interchange the variables.

$x = 5 y^{\frac{1}{5}} + 3$

Solve for $y$ .

Rewrite the equation as $5 y^{\frac{1}{5}} + 3 = x$ .

$5 y^{\frac{1}{5}} + 3 = x$

Move all terms not containing $y$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$5 y^{\frac{1}{5}} - x = - 3$

Add $x$ to both sides of the equation.

$5 y^{\frac{1}{5}} = - 3 + x$

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(5 y^{\frac{1}{5}})}^{5} = {(- 3 + x)}^{5}$

Simplify the exponent.

Simplify the left side.

Simplify ${(5 y^{\frac{1}{5}})}^{5}$ .

Apply the product rule to $5 y^{\frac{1}{5}}$ .

$5^{5} {(y^{\frac{1}{5}})}^{5} = {(- 3 + x)}^{5}$

Raise $5$ to the power of $5$ .

$3125 {(y^{\frac{1}{5}})}^{5} = {(- 3 + x)}^{5}$

Multiply the exponents in ${(y^{\frac{1}{5}})}^{5}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3125 y^{\frac{1}{5} \cdot 5} = {(- 3 + x)}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$3125 y^{\frac{1}{5} \cdot 5} = {(- 3 + x)}^{5}$

Rewrite the expression.

$3125 y^{1} = {(- 3 + x)}^{5}$

Simplify.

$3125 y = {(- 3 + x)}^{5}$

Simplify the right side.

Simplify ${(- 3 + x)}^{5}$ .

Use the Binomial Theorem.

$3125 y = {(- 3)}^{5} + 5 {(- 3)}^{4} x + 10 {(- 3)}^{3} x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Simplify each term.

Raise $- 3$ to the power of $5$ .

$3125 y = - 243 + 5 {(- 3)}^{4} x + 10 {(- 3)}^{3} x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Raise $- 3$ to the power of $4$ .

$3125 y = - 243 + 5 \cdot 81 x + 10 {(- 3)}^{3} x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Multiply $5$ by $81$ .

$3125 y = - 243 + 405 x + 10 {(- 3)}^{3} x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Raise $- 3$ to the power of $3$ .

$3125 y = - 243 + 405 x + 10 \cdot - 27 x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Multiply $10$ by $- 27$ .

$3125 y = - 243 + 405 x - 270 x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Raise $- 3$ to the power of $2$ .

$3125 y = - 243 + 405 x - 270 x^{2} + 10 \cdot 9 x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Multiply $10$ by $9$ .

$3125 y = - 243 + 405 x - 270 x^{2} + 90 x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Multiply $5$ by $- 3$ .

$3125 y = - 243 + 405 x - 270 x^{2} + 90 x^{3} - 15 x^{4} + x^{5}$

Divide each term in $3125 y = - 243 + 405 x - 270 x^{2} + 90 x^{3} - 15 x^{4} + x^{5}$ by $3125$ and simplify.

Divide each term in $3125 y = - 243 + 405 x - 270 x^{2} + 90 x^{3} - 15 x^{4} + x^{5}$ by $3125$ .

$\frac{3125 y}{3125} = \frac{- 243}{3125} + \frac{405 x}{3125} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Simplify the left side.

Cancel the common factor of $3125$ .

Cancel the common factor.

$\frac{3125 y}{3125} = \frac{- 243}{3125} + \frac{405 x}{3125} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Divide $y$ by $1$ .

$y = \frac{- 243}{3125} + \frac{405 x}{3125} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{243}{3125} + \frac{405 x}{3125} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Cancel the common factor of $405$ and $3125$ .

Factor $5$ out of $405 x$ .

$y = - \frac{243}{3125} + \frac{5 (81 x)}{3125} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Cancel the common factors.

Factor $5$ out of $3125$ .

$y = - \frac{243}{3125} + \frac{5 (81 x)}{5 (625)} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Cancel the common factor.

$y = - \frac{243}{3125} + \frac{5 (81 x)}{5 \cdot 625} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Rewrite the expression.

$y = - \frac{243}{3125} + \frac{81 x}{625} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Cancel the common factor of $- 270$ and $3125$ .

Factor $5$ out of $- 270 x^{2}$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} + \frac{5 (- 54 x^{2})}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Cancel the common factors.

Factor $5$ out of $3125$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} + \frac{5 (- 54 x^{2})}{5 (625)} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Cancel the common factor.

$y = - \frac{243}{3125} + \frac{81 x}{625} + \frac{5 (- 54 x^{2})}{5 \cdot 625} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Rewrite the expression.

$y = - \frac{243}{3125} + \frac{81 x}{625} + \frac{- 54 x^{2}}{625} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Move the negative in front of the fraction.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{(54) x^{2}}{625} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Cancel the common factor of $90$ and $3125$ .

Factor $5$ out of $90 x^{3}$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{5 (18 x^{3})}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Cancel the common factors.

Factor $5$ out of $3125$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{5 (18 x^{3})}{5 (625)} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Cancel the common factor.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{5 (18 x^{3})}{5 \cdot 625} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Rewrite the expression.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Cancel the common factor of $- 15$ and $3125$ .

Factor $5$ out of $- 15 x^{4}$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} + \frac{5 (- 3 x^{4})}{3125} + \frac{x^{5}}{3125}$

Cancel the common factors.

Factor $5$ out of $3125$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} + \frac{5 (- 3 x^{4})}{5 (625)} + \frac{x^{5}}{3125}$

Cancel the common factor.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} + \frac{5 (- 3 x^{4})}{5 \cdot 625} + \frac{x^{5}}{3125}$

Rewrite the expression.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} + \frac{- 3 x^{4}}{625} + \frac{x^{5}}{3125}$

Move the negative in front of the fraction.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}$

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}$

Verify if $f^{- 1} (x) = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}$ is the inverse of $f (x) = 5 x^{\frac{1}{5}} + 3$ .

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} (5 x^{\frac{1}{5}} + 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{81 (5 x^{\frac{1}{5}} + 3)}{625} - \frac{54 {(5 x^{\frac{1}{5}} + 3)}^{2}}{625} + \frac{18 {(5 x^{\frac{1}{5}} + 3)}^{3}}{625} - \frac{3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125}$

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{81 (5 x^{\frac{1}{5}} + 3) - 54 {(5 x^{\frac{1}{5}} + 3)}^{2} + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Simplify each term.

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{81 (5 x^{\frac{1}{5}}) + 81 \cdot 3 - 54 {(5 x^{\frac{1}{5}} + 3)}^{2} + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $5$ by $81$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 81 \cdot 3 - 54 {(5 x^{\frac{1}{5}} + 3)}^{2} + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $81$ by $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 {(5 x^{\frac{1}{5}} + 3)}^{2} + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Rewrite ${(5 x^{\frac{1}{5}} + 3)}^{2}$ as $(5 x^{\frac{1}{5}} + 3) (5 x^{\frac{1}{5}} + 3)$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 ((5 x^{\frac{1}{5}} + 3) (5 x^{\frac{1}{5}} + 3)) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Expand $(5 x^{\frac{1}{5}} + 3) (5 x^{\frac{1}{5}} + 3)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 x^{\frac{1}{5}} (5 x^{\frac{1}{5}} + 3) + 3 (5 x^{\frac{1}{5}} + 3)) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 x^{\frac{1}{5}} (5 x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}} + 3)) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 x^{\frac{1}{5}} (5 x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 \cdot (5 x^{\frac{1}{5}} x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

Move $x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 \cdot (5 (x^{\frac{1}{5}} x^{\frac{1}{5}})) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 \cdot (5 x^{\frac{1}{5} + \frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 \cdot (5 x^{\frac{1 + 1}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Add $1$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 \cdot (5 x^{\frac{2}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $5$ by $5$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $3$ by $5$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}} + 15 x^{\frac{1}{5}} + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $5$ by $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}} + 15 x^{\frac{1}{5}} + 15 x^{\frac{1}{5}} + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $3$ by $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}} + 15 x^{\frac{1}{5}} + 15 x^{\frac{1}{5}} + 9) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Add $15 x^{\frac{1}{5}}$ and $15 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} + 9) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}}) - 54 (30 x^{\frac{1}{5}}) - 54 \cdot 9 + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Simplify.

Multiply $25$ by $- 54$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 54 (30 x^{\frac{1}{5}}) - 54 \cdot 9 + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $30$ by $- 54$ .

Multiply $- 54$ by $9$ .

Use the Binomial Theorem.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 ({(5 x^{\frac{1}{5}})}^{3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Simplify each term.

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (5^{3} {(x^{\frac{1}{5}})}^{3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Raise $5$ to the power of $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 {(x^{\frac{1}{5}})}^{3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply the exponents in ${(x^{\frac{1}{5}})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{1}{5} \cdot 3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Combine $\frac{1}{5}$ and $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 3 (5^{2} {(x^{\frac{1}{5}})}^{2}) \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Raise $5$ to the power of $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 3 (25 {(x^{\frac{1}{5}})}^{2}) \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply the exponents in ${(x^{\frac{1}{5}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 3 (25 x^{\frac{1}{5} \cdot 2}) \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Combine $\frac{1}{5}$ and $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 3 (25 x^{\frac{2}{5}}) \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $25$ by $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 75 x^{\frac{2}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $3$ by $75$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $3$ by $3^{2}$ by adding the exponents.

Move $3^{2}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 3^{2} \cdot (3 (5 x^{\frac{1}{5}})) + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $3^{2}$ by $3$ .

Raise $3$ to the power of $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 3^{2} \cdot (3 (5 x^{\frac{1}{5}})) + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 3^{2 + 1} (5 x^{\frac{1}{5}}) + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Add $2$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 3^{3} (5 x^{\frac{1}{5}}) + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Raise $3$ to the power of $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 27 (5 x^{\frac{1}{5}}) + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $5$ by $27$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 135 x^{\frac{1}{5}} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Raise $3$ to the power of $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 135 x^{\frac{1}{5}} + 27) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}}) + 18 (225 x^{\frac{2}{5}}) + 18 (135 x^{\frac{1}{5}}) + 18 \cdot 27 - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Simplify.

Multiply $125$ by $18$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 18 (225 x^{\frac{2}{5}}) + 18 (135 x^{\frac{1}{5}}) + 18 \cdot 27 - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $225$ by $18$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 18 (135 x^{\frac{1}{5}}) + 18 \cdot 27 - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $135$ by $18$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 18 \cdot 27 - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Multiply $18$ by $27$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Use the Binomial Theorem.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 ({(5 x^{\frac{1}{5}})}^{4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Simplify each term.

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (5^{4} {(x^{\frac{1}{5}})}^{4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Raise $5$ to the power of $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 {(x^{\frac{1}{5}})}^{4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Multiply the exponents in ${(x^{\frac{1}{5}})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{1}{5} \cdot 4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Combine $\frac{1}{5}$ and $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 4 (5^{3} {(x^{\frac{1}{5}})}^{3}) \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Raise $5$ to the power of $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 4 (125 {(x^{\frac{1}{5}})}^{3}) \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Multiply the exponents in ${(x^{\frac{1}{5}})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 4 (125 x^{\frac{1}{5} \cdot 3}) \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Combine $\frac{1}{5}$ and $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 4 (125 x^{\frac{3}{5}}) \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Multiply $125$ by $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 500 x^{\frac{3}{5}} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Multiply $3$ by $500$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 6 (5^{2} {(x^{\frac{1}{5}})}^{2}) \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Raise $5$ to the power of $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 6 (25 {(x^{\frac{1}{5}})}^{2}) \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Multiply the exponents in ${(x^{\frac{1}{5}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 6 (25 x^{\frac{1}{5} \cdot 2}) \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Combine $\frac{1}{5}$ and $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 6 (25 x^{\frac{2}{5}}) \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Multiply $25$ by $6$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Raise $3$ to the power of $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} \cdot 9 + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Multiply $9$ by $150$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 1350 x^{\frac{2}{5}} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Multiply $5$ by $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 1350 x^{\frac{2}{5}} + 20 x^{\frac{1}{5}} \cdot 3^{3} + 3^{4})}{625}$

Raise $3$ to the power of $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 1350 x^{\frac{2}{5}} + 20 x^{\frac{1}{5}} \cdot 27 + 3^{4})}{625}$

Multiply $27$ by $20$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 1350 x^{\frac{2}{5}} + 540 x^{\frac{1}{5}} + 3^{4})}{625}$

Raise $3$ to the power of $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 1350 x^{\frac{2}{5}} + 540 x^{\frac{1}{5}} + 81)}{625}$

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}}) - 3 (1500 x^{\frac{3}{5}}) - 3 (1350 x^{\frac{2}{5}}) - 3 (540 x^{\frac{1}{5}}) - 3 \cdot 81}{625}$

Simplify.

Multiply $625$ by $- 3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 3 (1500 x^{\frac{3}{5}}) - 3 (1350 x^{\frac{2}{5}}) - 3 (540 x^{\frac{1}{5}}) - 3 \cdot 81}{625}$

Multiply $1500$ by $- 3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 3 (1350 x^{\frac{2}{5}}) - 3 (540 x^{\frac{1}{5}}) - 3 \cdot 81}{625}$

Multiply $1350$ by $- 3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 3 (540 x^{\frac{1}{5}}) - 3 \cdot 81}{625}$

Multiply $540$ by $- 3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 3 \cdot 81}{625}$

Multiply $- 3$ by $81$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 243}{625}$

Simplify by adding terms.

Combine the opposite terms in $405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 243$ .

Add $- 486$ and $486$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 0 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 243}{625}$

Add $405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}}$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 243}{625}$

Subtract $4050 x^{\frac{2}{5}}$ from $4050 x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} + 0 - 1620 x^{\frac{1}{5}} - 243}{625}$

Add $405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}}$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 1620 x^{\frac{1}{5}} - 243}{625}$

Subtract $243$ from $243$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 1620 x^{\frac{1}{5}} + 0}{625}$

Add $405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 1620 x^{\frac{1}{5}}$ and $0$ .

Subtract $1620 x^{\frac{1}{5}}$ from $405 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 1215 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 1620 x^{\frac{1}{5}}}{625}$

Add $- 1215 x^{\frac{1}{5}}$ and $2430 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{1215 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} + 2250 x^{\frac{3}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 1620 x^{\frac{1}{5}}}{625}$

Subtract $1620 x^{\frac{1}{5}}$ from $1215 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} + 2250 x^{\frac{3}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}}}{625}$

Subtract $4500 x^{\frac{3}{5}}$ from $2250 x^{\frac{3}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} - 1875 x^{\frac{4}{5}} - 2250 x^{\frac{3}{5}}}{625}$

Simplify each term.

Move the negative in front of the fraction.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} - 1875 x^{\frac{4}{5}} - 2250 x^{\frac{3}{5}}}{625}$

Factor $- 15 x^{\frac{1}{5}}$ out of $- 405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} - 1875 x^{\frac{4}{5}} - 2250 x^{\frac{3}{5}}$ .

Factor $- 15 x^{\frac{1}{5}}$ out of $- 405 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 15 x^{\frac{1}{5}} \cdot 27 - 1350 x^{\frac{2}{5}} - 1875 x^{\frac{4}{5}} - 2250 x^{\frac{3}{5}}}{625}$

Factor $- 15 x^{\frac{1}{5}}$ out of $- 1350 x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 15 x^{\frac{1}{5}} \cdot 27 - 15 x^{\frac{1}{5}} (90 x^{\frac{1}{5}}) - 1875 x^{\frac{4}{5}} - 2250 x^{\frac{3}{5}}}{625}$

Factor $- 15 x^{\frac{1}{5}}$ out of $- 1875 x^{\frac{4}{5}}$ .

Factor $- 15 x^{\frac{1}{5}}$ out of $- 2250 x^{\frac{3}{5}}$ .

Factor $- 15 x^{\frac{1}{5}}$ out of $- 15 x^{\frac{1}{5}} (27) - 15 x^{\frac{1}{5}} (90 x^{\frac{1}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 15 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}}) - 15 x^{\frac{1}{5}} (125 x^{\frac{3}{5}}) - 15 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{625}$

Factor $- 15 x^{\frac{1}{5}}$ out of $- 15 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}}) - 15 x^{\frac{1}{5}} (125 x^{\frac{3}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 15 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}}) - 15 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{625}$

Factor $- 15 x^{\frac{1}{5}}$ out of $- 15 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}}) - 15 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})$ .

Factor $5$ out of $- 15 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{5 (- 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}}))}{625}$

Cancel the common factors.

Factor $5$ out of $625$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Move the negative in front of the fraction.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + {(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Simplify each term.

Simplify the numerator.

Use the Binomial Theorem.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + {(5 x^{\frac{1}{5}})}^{5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Simplify each term.

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 5^{5} {(x^{\frac{1}{5}})}^{5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Raise $5$ to the power of $5$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 {(x^{\frac{1}{5}})}^{5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x^{\frac{1}{5} \cdot 5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Cancel the common factor of $5$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Simplify.

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 5 (5^{4} {(x^{\frac{1}{5}})}^{4}) \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $5$ by $5^{4}$ by adding the exponents.

Move $5^{4}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 5^{4} \cdot (5 {(x^{\frac{1}{5}})}^{4}) \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $5^{4}$ by $5$ .

Raise $5$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 5^{4 + 1} {(x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Add $4$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 5^{5} {(x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Raise $5$ to the power of $5$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 3125 {(x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply the exponents in ${(x^{\frac{1}{5}})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 3125 x^{\frac{1}{5} \cdot 4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Combine $\frac{1}{5}$ and $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 3125 x^{\frac{4}{5}} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $3$ by $3125$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 10 (5^{3} {(x^{\frac{1}{5}})}^{3}) \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Raise $5$ to the power of $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 10 (125 {(x^{\frac{1}{5}})}^{3}) \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply the exponents in ${(x^{\frac{1}{5}})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 10 (125 x^{\frac{1}{5} \cdot 3}) \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Combine $\frac{1}{5}$ and $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 10 (125 x^{\frac{3}{5}}) \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $125$ by $10$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Raise $3$ to the power of $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} \cdot 9 + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $9$ by $1250$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 10 (5^{2} {(x^{\frac{1}{5}})}^{2}) \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Raise $5$ to the power of $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 10 (25 {(x^{\frac{1}{5}})}^{2}) \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply the exponents in ${(x^{\frac{1}{5}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 10 (25 x^{\frac{1}{5} \cdot 2}) \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Combine $\frac{1}{5}$ and $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 10 (25 x^{\frac{2}{5}}) \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $25$ by $10$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 250 x^{\frac{2}{5}} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Raise $3$ to the power of $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 250 x^{\frac{2}{5}} \cdot 27 + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $27$ by $250$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $5$ by $5$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 25 x^{\frac{1}{5}} \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Raise $3$ to the power of $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 25 x^{\frac{1}{5}} \cdot 81 + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $81$ by $25$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Raise $3$ to the power of $5$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}} + 243}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Add $- 243$ and $243$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}} + 0}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Add $3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Reorder terms.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{3125 x + 11250 x^{\frac{3}{5}} + 9375 x^{\frac{4}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Rewrite $3125 x + 11250 x^{\frac{3}{5}} + 9375 x^{\frac{4}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}$ in a factored form.

Factor $25 x^{\frac{1}{5}}$ out of $3125 x + 11250 x^{\frac{3}{5}} + 9375 x^{\frac{4}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}$ .

Factor $25 x^{\frac{1}{5}}$ out of $3125 x$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 11250 x^{\frac{3}{5}} + 9375 x^{\frac{4}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $11250 x^{\frac{3}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}}) + 9375 x^{\frac{4}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $9375 x^{\frac{4}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $6750 x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + 25 x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $2025 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + 25 x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + 25 x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (375 x^{\frac{3}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}}) + 25 x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}}) + 25 x^{\frac{1}{5}} (270 x^{\frac{1}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (81)$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Reorder terms.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Factor $25$ out of $25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81)$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 (x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81))}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Cancel the common factors.

Factor $25$ out of $3125$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 (x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81))}{25 \cdot 125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81)}{125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81) - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Simplify each term.

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{x^{\frac{1}{5}} (450 x^{\frac{2}{5}}) + x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} \cdot 81 - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Simplify.

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{1}{5}} x^{\frac{2}{5}} + x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} \cdot 81 - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{1}{5}} x^{\frac{2}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} \cdot 81 - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{1}{5}} x^{\frac{2}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} \cdot 81 - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{1}{5}} x^{\frac{2}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + x^{\frac{1}{5}} \cdot 81 - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Move $81$ to the left of $x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{1}{5}} x^{\frac{2}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Simplify each term.

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{2}{5}}$ by adding the exponents.

Move $x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 (x^{\frac{2}{5}} x^{\frac{1}{5}}) + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{2}{5} + \frac{1}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{2 + 1}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Add $2$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{3}{5}}$ by adding the exponents.

Move $x^{\frac{3}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 (x^{\frac{3}{5}} x^{\frac{1}{5}}) + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{3}{5} + \frac{1}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{3 + 1}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Add $3$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

Move $x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 (x^{\frac{1}{5}} x^{\frac{1}{5}}) + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{1}{5} + \frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{1 + 1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Add $1$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{4}{5}}$ by adding the exponents.

Move $x^{\frac{4}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 (x^{\frac{4}{5}} x^{\frac{1}{5}}) + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x^{\frac{4}{5} + \frac{1}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x^{\frac{4 + 1}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Add $4$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x^{\frac{5}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Divide $5$ by $5$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} \cdot 27 - 3 x^{\frac{1}{5}} (90 x^{\frac{1}{5}}) - 3 x^{\frac{1}{5}} (125 x^{\frac{3}{5}}) - 3 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{125}$

Simplify.

Multiply $27$ by $- 3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (90 x^{\frac{1}{5}}) - 3 x^{\frac{1}{5}} (125 x^{\frac{3}{5}}) - 3 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{125}$

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{1}{5}} x^{\frac{1}{5}}) - 3 x^{\frac{1}{5}} (125 x^{\frac{3}{5}}) - 3 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{125}$

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{1}{5}} x^{\frac{1}{5}}) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{125}$

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{1}{5}} x^{\frac{1}{5}}) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Simplify each term.

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

Move $x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 (x^{\frac{1}{5}} x^{\frac{1}{5}})) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{1}{5} + \frac{1}{5}}) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{1 + 1}{5}}) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Add $1$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{2}{5}}) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Multiply $- 3$ by $90$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{3}{5}}$ by adding the exponents.

Move $x^{\frac{3}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 3 \cdot (125 (x^{\frac{3}{5}} x^{\frac{1}{5}})) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 3 \cdot (125 x^{\frac{3}{5} + \frac{1}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 3 \cdot (125 x^{\frac{3 + 1}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Add $3$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 3 \cdot (125 x^{\frac{4}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Multiply $- 3$ by $125$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{2}{5}}$ by adding the exponents.

Move $x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 3 \cdot (150 (x^{\frac{2}{5}} x^{\frac{1}{5}}))}{125}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 3 \cdot (150 x^{\frac{2}{5} + \frac{1}{5}})}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 3 \cdot (150 x^{\frac{2 + 1}{5}})}{125}$

Add $2$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 3 \cdot (150 x^{\frac{3}{5}})}{125}$

Multiply $- 3$ by $150$ .

Simplify terms.

Combine the opposite terms in $450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 450 x^{\frac{3}{5}}$ .

Subtract $81 x^{\frac{1}{5}}$ from $81 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 0 - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 450 x^{\frac{3}{5}}}{125}$

Add $450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 450 x^{\frac{3}{5}}}{125}$

Subtract $270 x^{\frac{2}{5}}$ from $270 x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 125 x + 0 - 375 x^{\frac{4}{5}} - 450 x^{\frac{3}{5}}}{125}$

Add $450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 125 x$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 125 x - 375 x^{\frac{4}{5}} - 450 x^{\frac{3}{5}}}{125}$

Subtract $375 x^{\frac{4}{5}}$ from $375 x^{\frac{4}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 125 x + 0 - 450 x^{\frac{3}{5}}}{125}$

Add $450 x^{\frac{3}{5}} + 125 x$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 125 x - 450 x^{\frac{3}{5}}}{125}$

Subtract $450 x^{\frac{3}{5}}$ from $450 x^{\frac{3}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{125 x + 0}{125}$

Add $125 x$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{125 x}{125}$

Cancel the common factor of $125$ .

Cancel the common factor.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{125 x}{125}$

Divide $x$ by $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = x$

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

Evaluate $f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125})}^{\frac{1}{5}} + 3$

Move $- \frac{243}{3125}$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(\frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125} - \frac{243}{3125})}^{\frac{1}{5}} + 3$

Move $\frac{81 x}{625}$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(- \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125} + \frac{81 x}{625} - \frac{243}{3125})}^{\frac{1}{5}} + 3$

Move $- \frac{54 x^{2}}{625}$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(\frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125} - \frac{54 x^{2}}{625} + \frac{81 x}{625} - \frac{243}{3125})}^{\frac{1}{5}} + 3$

Move $\frac{18 x^{3}}{625}$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(- \frac{3 x^{4}}{625} + \frac{x^{5}}{3125} + \frac{18 x^{3}}{625} - \frac{54 x^{2}}{625} + \frac{81 x}{625} - \frac{243}{3125})}^{\frac{1}{5}} + 3$

Reorder $- \frac{3 x^{4}}{625}$ and $\frac{x^{5}}{3125}$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(\frac{x^{5}}{3125} - \frac{3 x^{4}}{625} + \frac{18 x^{3}}{625} - \frac{54 x^{2}}{625} + \frac{81 x}{625} - \frac{243}{3125})}^{\frac{1}{5}} + 3$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}$ is the inverse of $f (x) = 5 x^{\frac{1}{5}} + 3$ .

$f^{- 1} (x) = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}$

Find the Inverse 5x^(1/5)+3

Related Questions

Support LCMGCF.com by disabling your adblocker.