Find the Inverse 15+ cube root of x

$15 + \sqrt[3]{x}$

Step 1

Interchange the variables.

$x = 15 + \sqrt[3]{y}$

Step 2

Solve for $y$ .

Rewrite the equation as $15 + \sqrt[3]{y} = x$ .

$15 + \sqrt[3]{y} = x$

Subtract $15$ from both sides of the equation.

$\sqrt[3]{y} = x - 15$

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{y}}^{3} = {(x - 15)}^{3}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

${(y^{\frac{1}{3}})}^{3} = {(x - 15)}^{3}$

Simplify the left side.

Simplify ${(y^{\frac{1}{3}})}^{3}$ .

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

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

$y^{\frac{1}{3} \cdot 3} = {(x - 15)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(x - 15)}^{3}$

Rewrite the expression.

$y^{1} = {(x - 15)}^{3}$

Simplify.

$y = {(x - 15)}^{3}$

Simplify the right side.

Simplify ${(x - 15)}^{3}$ .

Use the Binomial Theorem.

$y = x^{3} + 3 x^{2} \cdot - 15 + 3 x {(- 15)}^{2} + {(- 15)}^{3}$

Simplify each term.

Multiply $- 15$ by $3$ .

$y = x^{3} - 45 x^{2} + 3 x {(- 15)}^{2} + {(- 15)}^{3}$

Raise $- 15$ to the power of $2$ .

$y = x^{3} - 45 x^{2} + 3 x \cdot 225 + {(- 15)}^{3}$

Multiply $225$ by $3$ .

$y = x^{3} - 45 x^{2} + 675 x + {(- 15)}^{3}$

Raise $- 15$ to the power of $3$ .

$y = x^{3} - 45 x^{2} + 675 x - 3375$

Step 3

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

$f^{- 1} (x) = x^{3} - 45 x^{2} + 675 x - 3375$

Step 4

Verify if $f^{- 1} (x) = x^{3} - 45 x^{2} + 675 x - 3375$ is the inverse of $f (x) = 15 + \sqrt[3]{x}$ .

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 $f^{- 1} (15 + \sqrt[3]{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (15 + \sqrt[3]{x}) = {(15 + \sqrt[3]{x})}^{3} - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

Simplify each term.

Use the Binomial Theorem.

$f^{- 1} (15 + \sqrt[3]{x}) = 15^{3} + 3 \cdot (15^{2} \sqrt[3]{x}) + 3 \cdot (15 {\sqrt[3]{x}}^{2}) + {\sqrt[3]{x}}^{3} - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

Simplify each term.

Raise $15$ to the power of $3$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 3 \cdot (15^{2} \sqrt[3]{x}) + 3 \cdot (15 {\sqrt[3]{x}}^{2}) + {\sqrt[3]{x}}^{3} - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

Raise $15$ to the power of $2$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 3 \cdot (225 \sqrt[3]{x}) + 3 \cdot (15 {\sqrt[3]{x}}^{2}) + {\sqrt[3]{x}}^{3} - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

Multiply $3$ by $225$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 3 \cdot (15 {\sqrt[3]{x}}^{2}) + {\sqrt[3]{x}}^{3} - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

Multiply $3$ by $15$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 {\sqrt[3]{x}}^{2} + {\sqrt[3]{x}}^{3} - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + {\sqrt[3]{x}}^{3} - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + {(x^{\frac{1}{3}})}^{3} - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

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

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x^{\frac{1}{3} \cdot 3} - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

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

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x^{\frac{3}{3}} - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x^{\frac{3}{3}} - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

Rewrite the expression.

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

Simplify.

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 {(15 + \sqrt[3]{x})}^{2} + 675 (15 + \sqrt[3]{x}) - 3375$

Rewrite ${(15 + \sqrt[3]{x})}^{2}$ as $(15 + \sqrt[3]{x}) (15 + \sqrt[3]{x})$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 ((15 + \sqrt[3]{x}) (15 + \sqrt[3]{x})) + 675 (15 + \sqrt[3]{x}) - 3375$

Expand $(15 + \sqrt[3]{x}) (15 + \sqrt[3]{x})$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 (15 (15 + \sqrt[3]{x}) + \sqrt[3]{x} (15 + \sqrt[3]{x})) + 675 (15 + \sqrt[3]{x}) - 3375$

Apply the distributive property.

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 (15 \cdot 15 + 15 \sqrt[3]{x} + \sqrt[3]{x} (15 + \sqrt[3]{x})) + 675 (15 + \sqrt[3]{x}) - 3375$

Apply the distributive property.

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 (15 \cdot 15 + 15 \sqrt[3]{x} + \sqrt[3]{x} \cdot 15 + \sqrt[3]{x} \sqrt[3]{x}) + 675 (15 + \sqrt[3]{x}) - 3375$

Simplify and combine like terms.

Simplify each term.

Multiply $15$ by $15$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 (225 + 15 \sqrt[3]{x} + \sqrt[3]{x} \cdot 15 + \sqrt[3]{x} \sqrt[3]{x}) + 675 (15 + \sqrt[3]{x}) - 3375$

Move $15$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 (225 + 15 \sqrt[3]{x} + 15 \sqrt[3]{x} + \sqrt[3]{x} \sqrt[3]{x}) + 675 (15 + \sqrt[3]{x}) - 3375$

Multiply $\sqrt[3]{x} \sqrt[3]{x}$ .

Raise $\sqrt[3]{x}$ to the power of $1$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 (225 + 15 \sqrt[3]{x} + 15 \sqrt[3]{x} + \sqrt[3]{x} \sqrt[3]{x}) + 675 (15 + \sqrt[3]{x}) - 3375$

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

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 (225 + 15 \sqrt[3]{x} + 15 \sqrt[3]{x} + {\sqrt[3]{x}}^{1 + 1}) + 675 (15 + \sqrt[3]{x}) - 3375$

Add $1$ and $1$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 (225 + 15 \sqrt[3]{x} + 15 \sqrt[3]{x} + {\sqrt[3]{x}}^{2}) + 675 (15 + \sqrt[3]{x}) - 3375$

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 (225 + 15 \sqrt[3]{x} + 15 \sqrt[3]{x} + \sqrt[3]{x^{2}}) + 675 (15 + \sqrt[3]{x}) - 3375$

Add $15 \sqrt[3]{x}$ and $15 \sqrt[3]{x}$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 (225 + 30 \sqrt[3]{x} + \sqrt[3]{x^{2}}) + 675 (15 + \sqrt[3]{x}) - 3375$

Apply the distributive property.

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 45 \cdot 225 - 45 (30 \sqrt[3]{x}) - 45 \sqrt[3]{x^{2}} + 675 (15 + \sqrt[3]{x}) - 3375$

Simplify.

Multiply $- 45$ by $225$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 10125 - 45 (30 \sqrt[3]{x}) - 45 \sqrt[3]{x^{2}} + 675 (15 + \sqrt[3]{x}) - 3375$

Multiply $30$ by $- 45$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 10125 - 1350 \sqrt[3]{x} - 45 \sqrt[3]{x^{2}} + 675 (15 + \sqrt[3]{x}) - 3375$

Apply the distributive property.

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 10125 - 1350 \sqrt[3]{x} - 45 \sqrt[3]{x^{2}} + 675 \cdot 15 + 675 \sqrt[3]{x} - 3375$

Multiply $675$ by $15$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 10125 - 1350 \sqrt[3]{x} - 45 \sqrt[3]{x^{2}} + 10125 + 675 \sqrt[3]{x} - 3375$

Simplify by adding terms.

Combine the opposite terms in $3375 + 675 \sqrt[3]{x} + 45 \sqrt[3]{x^{2}} + x - 10125 - 1350 \sqrt[3]{x} - 45 \sqrt[3]{x^{2}} + 10125 + 675 \sqrt[3]{x} - 3375$ .

Subtract $45 \sqrt[3]{x^{2}}$ from $45 \sqrt[3]{x^{2}}$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + 0 + x - 10125 - 1350 \sqrt[3]{x} + 10125 + 675 \sqrt[3]{x} - 3375$

Add $3375 + 675 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + x - 10125 - 1350 \sqrt[3]{x} + 10125 + 675 \sqrt[3]{x} - 3375$

Add $- 10125$ and $10125$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + x + 0 - 1350 \sqrt[3]{x} + 675 \sqrt[3]{x} - 3375$

Add $3375 + 675 \sqrt[3]{x} + x$ and $0$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 3375 + 675 \sqrt[3]{x} + x - 1350 \sqrt[3]{x} + 675 \sqrt[3]{x} - 3375$

Subtract $3375$ from $3375$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 0 + 675 \sqrt[3]{x} + x - 1350 \sqrt[3]{x} + 675 \sqrt[3]{x}$

Add $0$ and $675 \sqrt[3]{x}$ .

$f^{- 1} (15 + \sqrt[3]{x}) = 675 \sqrt[3]{x} + x - 1350 \sqrt[3]{x} + 675 \sqrt[3]{x}$

Subtract $1350 \sqrt[3]{x}$ from $675 \sqrt[3]{x}$ .

$f^{- 1} (15 + \sqrt[3]{x}) = x - 675 \sqrt[3]{x} + 675 \sqrt[3]{x}$

Combine the opposite terms in $x - 675 \sqrt[3]{x} + 675 \sqrt[3]{x}$ .

Add $- 675 \sqrt[3]{x}$ and $675 \sqrt[3]{x}$ .

$f^{- 1} (15 + \sqrt[3]{x}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (15 + \sqrt[3]{x}) = x$

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

Set up the composite result function.

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

Evaluate $f (x^{3} - 45 x^{2} + 675 x - 3375)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{3} - 45 x^{2} + 675 x - 3375) = 15 + \sqrt[3]{x^{3} - 45 x^{2} + 675 x - 3375}$

Simplify each term.

Make each term match the terms from the binomial theorem formula.

$f (x^{3} - 45 x^{2} + 675 x - 3375) = 15 + \sqrt[3]{x^{3} - 3 \cdot (15 x^{2}) + 3 \cdot (15^{2} x) - 1 \cdot 15^{3}}$

Factor $x^{3} - 3 \cdot 15 x^{2} + 3 \cdot 15^{2} x - 1 \cdot 15^{3}$ using the binomial theorem.

$f (x^{3} - 45 x^{2} + 675 x - 3375) = 15 + \sqrt[3]{{(x - 15)}^{3}}$

Pull terms out from under the radical, assuming real numbers.

$f (x^{3} - 45 x^{2} + 675 x - 3375) = 15 + x - 15$

Combine the opposite terms in $15 + x - 15$ .

Subtract $15$ from $15$ .

$f (x^{3} - 45 x^{2} + 675 x - 3375) = x + 0$

Add $x$ and $0$ .

$f (x^{3} - 45 x^{2} + 675 x - 3375) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{3} - 45 x^{2} + 675 x - 3375$ is the inverse of $f (x) = 15 + \sqrt[3]{x}$ .

$f^{- 1} (x) = x^{3} - 45 x^{2} + 675 x - 3375$

Find the Inverse 15+ cube root of x

Related Questions

Support LCMGCF.com by disabling your adblocker.