Find the Inverse f(x)=8((x+10)/9)^(1/7)

$f (x) = 8 {(\frac{x + 10}{9})}^{\frac{1}{7}}$

Step 1

Write $f (x) = 8 {(\frac{x + 10}{9})}^{\frac{1}{7}}$ as an equation.

$y = 8 {(\frac{x + 10}{9})}^{\frac{1}{7}}$

Step 2

Interchange the variables.

$x = 8 {(\frac{y + 10}{9})}^{\frac{1}{7}}$

Step 3

Solve for $y$ .

Rewrite the equation as $8 {(\frac{y + 10}{9})}^{\frac{1}{7}} = x$ .

$8 {(\frac{y + 10}{9})}^{\frac{1}{7}} = x$

Divide each term in $8 {(\frac{y + 10}{9})}^{\frac{1}{7}} = x$ by $8$ and simplify.

Divide each term in $8 {(\frac{y + 10}{9})}^{\frac{1}{7}} = x$ by $8$ .

$\frac{8 {(\frac{y + 10}{9})}^{\frac{1}{7}}}{8} = \frac{x}{8}$

Simplify the left side.

Cancel the common factor.

$\frac{8 {(\frac{y + 10}{9})}^{\frac{1}{7}}}{8} = \frac{x}{8}$

Simplify the expression.

Divide ${(\frac{y + 10}{9})}^{\frac{1}{7}}$ by $1$ .

${(\frac{y + 10}{9})}^{\frac{1}{7}} = \frac{x}{8}$

Apply the product rule to $\frac{y + 10}{9}$ .

$\frac{{(y + 10)}^{\frac{1}{7}}}{9^{\frac{1}{7}}} = \frac{x}{8}$

Multiply both sides of the equation by $9^{\frac{1}{7}}$ .

$9^{\frac{1}{7}} \frac{{(y + 10)}^{\frac{1}{7}}}{9^{\frac{1}{7}}} = 9^{\frac{1}{7}} \frac{x}{8}$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $9^{\frac{1}{7}}$ .

Cancel the common factor.

$9^{\frac{1}{7}} \frac{{(y + 10)}^{\frac{1}{7}}}{9^{\frac{1}{7}}} = 9^{\frac{1}{7}} \frac{x}{8}$

Rewrite the expression.

${(y + 10)}^{\frac{1}{7}} = 9^{\frac{1}{7}} \frac{x}{8}$

Simplify the right side.

Simplify $9^{\frac{1}{7}} \frac{x}{8}$ .

Combine $9^{\frac{1}{7}}$ and $\frac{x}{8}$ .

${(y + 10)}^{\frac{1}{7}} = \frac{9^{\frac{1}{7}} x}{8}$

Reorder factors in $\frac{9^{\frac{1}{7}} x}{8}$ .

${(y + 10)}^{\frac{1}{7}} = \frac{x \cdot 9^{\frac{1}{7}}}{8}$

Raise each side of the equation to the power of $7$ to eliminate the fractional exponent on the left side.

${({(y + 10)}^{\frac{1}{7}})}^{7} = {(\frac{x \cdot 9^{\frac{1}{7}}}{8})}^{7}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(y + 10)}^{\frac{1}{7}})}^{7}$ .

Multiply the exponents in ${({(y + 10)}^{\frac{1}{7}})}^{7}$ .

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

${(y + 10)}^{\frac{1}{7} \cdot 7} = {(\frac{x \cdot 9^{\frac{1}{7}}}{8})}^{7}$

Cancel the common factor of $7$ .

Cancel the common factor.

${(y + 10)}^{\frac{1}{7} \cdot 7} = {(\frac{x \cdot 9^{\frac{1}{7}}}{8})}^{7}$

Rewrite the expression.

${(y + 10)}^{1} = {(\frac{x \cdot 9^{\frac{1}{7}}}{8})}^{7}$

Simplify.

$y + 10 = {(\frac{x \cdot 9^{\frac{1}{7}}}{8})}^{7}$

Simplify the right side.

Simplify ${(\frac{x \cdot 9^{\frac{1}{7}}}{8})}^{7}$ .

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{x \cdot 9^{\frac{1}{7}}}{8}$ .

$y + 10 = \frac{{(x \cdot 9^{\frac{1}{7}})}^{7}}{8^{7}}$

Apply the product rule to $x \cdot 9^{\frac{1}{7}}$ .

$y + 10 = \frac{x^{7} \cdot {(9^{\frac{1}{7}})}^{7}}{8^{7}}$

Simplify the numerator.

Multiply the exponents in ${(9^{\frac{1}{7}})}^{7}$ .

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

$y + 10 = \frac{x^{7} \cdot 9^{\frac{1}{7} \cdot 7}}{8^{7}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$y + 10 = \frac{x^{7} \cdot 9^{\frac{1}{7} \cdot 7}}{8^{7}}$

Rewrite the expression.

$y + 10 = \frac{x^{7} \cdot 9^{1}}{8^{7}}$

Evaluate the exponent.

$y + 10 = \frac{x^{7} \cdot 9}{8^{7}}$

Simplify the expression.

Raise $8$ to the power of $7$ .

$y + 10 = \frac{x^{7} \cdot 9}{2097152}$

Move $9$ to the left of $x^{7}$ .

$y + 10 = \frac{9 x^{7}}{2097152}$

Subtract $10$ from both sides of the equation.

$y = \frac{9 x^{7}}{2097152} - 10$

Step 4

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

$f^{- 1} (x) = \frac{9 x^{7}}{2097152} - 10$

Step 5

Verify if $f^{- 1} (x) = \frac{9 x^{7}}{2097152} - 10$ is the inverse of $f (x) = 8 {(\frac{x + 10}{9})}^{\frac{1}{7}}$ .

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 $f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{9 {(8 {(\frac{x + 10}{9})}^{\frac{1}{7}})}^{7}}{2097152} - 10$

Simplify each term.

Simplify the numerator.

Apply the product rule to $8 {(\frac{x + 10}{9})}^{\frac{1}{7}}$ .

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{9 \cdot (8^{7} {({(\frac{x + 10}{9})}^{\frac{1}{7}})}^{7})}{2097152} - 10$

Raise $8$ to the power of $7$ .

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{9 \cdot (2097152 {({(\frac{x + 10}{9})}^{\frac{1}{7}})}^{7})}{2097152} - 10$

Multiply the exponents in ${({(\frac{x + 10}{9})}^{\frac{1}{7}})}^{7}$ .

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

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{9 \cdot (2097152 {(\frac{x + 10}{9})}^{\frac{1}{7} \cdot 7})}{2097152} - 10$

Cancel the common factor of $7$ .

Cancel the common factor.

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{9 \cdot (2097152 {(\frac{x + 10}{9})}^{\frac{1}{7} \cdot 7})}{2097152} - 10$

Rewrite the expression.

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{9 \cdot (2097152 (\frac{x + 10}{9}))}{2097152} - 10$

Simplify.

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{9 \cdot (2097152 (\frac{x + 10}{9}))}{2097152} - 10$

Combine exponents.

Multiply $9$ by $2097152$ .

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{18874368 (\frac{x + 10}{9})}{2097152} - 10$

Combine $18874368$ and $\frac{x + 10}{9}$ .

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{\frac{18874368 (x + 10)}{9}}{2097152} - 10$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{18874368 (x + 10)}{9}$ by cancelling the common factors.

Factor $9$ out of $18874368 (x + 10)$ .

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{\frac{9 (2097152 (x + 10))}{9}}{2097152} - 10$

Factor $9$ out of $9$ .

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{\frac{9 (2097152 (x + 10))}{9 (1)}}{2097152} - 10$

Cancel the common factor.

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{\frac{9 (2097152 (x + 10))}{9 \cdot 1}}{2097152} - 10$

Rewrite the expression.

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{\frac{2097152 (x + 10)}{1}}{2097152} - 10$

Divide $2097152 (x + 10)$ by $1$ .

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{2097152 (x + 10)}{2097152} - 10$

Cancel the common factor of $2097152$ .

Cancel the common factor.

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = \frac{2097152 (x + 10)}{2097152} - 10$

Divide $x + 10$ by $1$ .

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = x + 10 - 10$

Combine the opposite terms in $x + 10 - 10$ .

Subtract $10$ from $10$ .

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (8 {(\frac{x + 10}{9})}^{\frac{1}{7}}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{9 x^{7}}{2097152} - 10)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{9 x^{7}}{2097152} - 10) = 8 {(\frac{(\frac{9 x^{7}}{2097152} - 10) + 10}{9})}^{\frac{1}{7}}$

Simplify the numerator.

Add $- 10$ and $10$ .

$f (\frac{9 x^{7}}{2097152} - 10) = 8 {(\frac{\frac{9 x^{7}}{2097152} + 0}{9})}^{\frac{1}{7}}$

Add $\frac{9 x^{7}}{2097152}$ and $0$ .

$f (\frac{9 x^{7}}{2097152} - 10) = 8 {(\frac{\frac{9 x^{7}}{2097152}}{9})}^{\frac{1}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{9 x^{7}}{2097152} - 10) = 8 {(\frac{9 x^{7}}{2097152} \cdot \frac{1}{9})}^{\frac{1}{7}}$

Simplify terms.

Combine.

$f (\frac{9 x^{7}}{2097152} - 10) = 8 {(\frac{9 x^{7} \cdot 1}{2097152 \cdot 9})}^{\frac{1}{7}}$

Cancel the common factor of $9$ .

Cancel the common factor.

$f (\frac{9 x^{7}}{2097152} - 10) = 8 {(\frac{9 x^{7} \cdot 1}{2097152 \cdot 9})}^{\frac{1}{7}}$

Rewrite the expression.

$f (\frac{9 x^{7}}{2097152} - 10) = 8 {(\frac{x^{7} \cdot 1}{2097152})}^{\frac{1}{7}}$

Simplify the expression.

Multiply $x^{7}$ by $1$ .

$f (\frac{9 x^{7}}{2097152} - 10) = 8 {(\frac{x^{7}}{2097152})}^{\frac{1}{7}}$

Apply the product rule to $\frac{x^{7}}{2097152}$ .

$f (\frac{9 x^{7}}{2097152} - 10) = 8 (\frac{{(x^{7})}^{\frac{1}{7}}}{2097152^{\frac{1}{7}}})$

Simplify the numerator.

Multiply the exponents in ${(x^{7})}^{\frac{1}{7}}$ .

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

$f (\frac{9 x^{7}}{2097152} - 10) = 8 (\frac{x^{7 (\frac{1}{7})}}{2097152^{\frac{1}{7}}})$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\frac{9 x^{7}}{2097152} - 10) = 8 (\frac{x^{7 (\frac{1}{7})}}{2097152^{\frac{1}{7}}})$

Rewrite the expression.

$f (\frac{9 x^{7}}{2097152} - 10) = 8 (\frac{x}{2097152^{\frac{1}{7}}})$

Simplify.

$f (\frac{9 x^{7}}{2097152} - 10) = 8 (\frac{x}{2097152^{\frac{1}{7}}})$

Simplify the denominator.

Rewrite $2097152$ as $8^{7}$ .

$f (\frac{9 x^{7}}{2097152} - 10) = 8 (\frac{x}{{(8^{7})}^{\frac{1}{7}}})$

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

$f (\frac{9 x^{7}}{2097152} - 10) = 8 (\frac{x}{8^{7 (\frac{1}{7})}})$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\frac{9 x^{7}}{2097152} - 10) = 8 (\frac{x}{8^{7 (\frac{1}{7})}})$

Rewrite the expression.

$f (\frac{9 x^{7}}{2097152} - 10) = 8 (\frac{x}{8})$

Evaluate the exponent.

$f (\frac{9 x^{7}}{2097152} - 10) = 8 (\frac{x}{8})$

Cancel the common factor of $8$ .

Cancel the common factor.

$f (\frac{9 x^{7}}{2097152} - 10) = 8 (\frac{x}{8})$

Rewrite the expression.

$f (\frac{9 x^{7}}{2097152} - 10) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{9 x^{7}}{2097152} - 10$ is the inverse of $f (x) = 8 {(\frac{x + 10}{9})}^{\frac{1}{7}}$ .

$f^{- 1} (x) = \frac{9 x^{7}}{2097152} - 10$

Find the Inverse f(x)=8((x+10)/9)^(1/7)

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