Algebra Examples

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

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 $y$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Simplify the expression.

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Step 3.2.2.2.1

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

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

Step 3.2.2.2.2

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

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

Step 3.3

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

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

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

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

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Step 3.4.1.1.1

Cancel the common factor.

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

Step 3.4.1.1.2

Rewrite the expression.

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

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

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

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

Step 3.4.2.1.2

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

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

Step 3.5

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

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

Step 3.6

Simplify the exponent.

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Step 3.6.1

Simplify the left side.

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Step 3.6.1.1

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

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Step 3.6.1.1.1

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

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Step 3.6.1.1.1.1

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

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

Step 3.6.1.1.1.2

Cancel the common factor of $7$ .

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Step 3.6.1.1.1.2.1

Cancel the common factor.

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

Step 3.6.1.1.1.2.2

Rewrite the expression.

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

Step 3.6.1.1.2

Simplify.

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

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

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

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Step 3.6.2.1.1

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

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Step 3.6.2.1.1.1

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

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

Step 3.6.2.1.1.2

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

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

Step 3.6.2.1.2

Simplify the numerator.

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Step 3.6.2.1.2.1

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

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Step 3.6.2.1.2.1.1

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

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

Step 3.6.2.1.2.1.2

Cancel the common factor of $7$ .

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Step 3.6.2.1.2.1.2.1

Cancel the common factor.

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

Step 3.6.2.1.2.1.2.2

Rewrite the expression.

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

Step 3.6.2.1.2.2

Evaluate the exponent.

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

Step 3.6.2.1.3

Simplify the expression.

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Step 3.6.2.1.3.1

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

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

Step 3.6.2.1.3.2

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

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

Step 3.7

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

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

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

Step 5.2.3.1.2

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

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

Step 5.2.3.1.3

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

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Step 5.2.3.1.3.1

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

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

Step 5.2.3.1.3.2

Cancel the common factor of $7$ .

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Step 5.2.3.1.3.2.1

Cancel the common factor.

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

Step 5.2.3.1.3.2.2

Rewrite the expression.

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

Step 5.2.3.1.4

Simplify.

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

Step 5.2.3.1.5

Combine exponents.

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Step 5.2.3.1.5.1

Multiply $9$ by $2097152$ .

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

Step 5.2.3.1.5.2

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

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

Step 5.2.3.1.6

Reduce the expression by cancelling the common factors.

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Step 5.2.3.1.6.1

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

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Step 5.2.3.1.6.1.1

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

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

Step 5.2.3.1.6.1.2

Factor $9$ out of $9$ .

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

Step 5.2.3.1.6.1.3

Cancel the common factor.

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

Step 5.2.3.1.6.1.4

Rewrite the expression.

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

Step 5.2.3.1.6.2

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

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

Step 5.2.3.2

Cancel the common factor of $2097152$ .

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Step 5.2.3.2.1

Cancel the common factor.

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

Step 5.2.3.2.2

Divide $x + 10$ by $1$ .

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

Step 5.2.4

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

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Step 5.2.4.1

Subtract $10$ from $10$ .

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

Step 5.2.4.2

Add $x$ and $0$ .

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Add $- 10$ and $10$ .

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

Step 5.3.3.2

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

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

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.5

Simplify terms.

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Step 5.3.5.1

Combine.

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

Step 5.3.5.2

Cancel the common factor of $9$ .

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Step 5.3.5.2.1

Cancel the common factor.

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

Step 5.3.5.2.2

Rewrite the expression.

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

Step 5.3.5.3

Simplify the expression.

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Step 5.3.5.3.1

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

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

Step 5.3.5.3.2

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

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

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

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

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Step 5.3.6.1.1

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

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

Step 5.3.6.1.2

Cancel the common factor of $7$ .

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Step 5.3.6.1.2.1

Cancel the common factor.

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

Step 5.3.6.1.2.2

Rewrite the expression.

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

Step 5.3.6.2

Simplify.

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

Step 5.3.7

Simplify the denominator.

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Step 5.3.7.1

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

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

Step 5.3.7.2

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

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

Step 5.3.7.3

Cancel the common factor of $7$ .

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Step 5.3.7.3.1

Cancel the common factor.

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

Step 5.3.7.3.2

Rewrite the expression.

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

Step 5.3.7.4

Evaluate the exponent.

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

Step 5.3.8

Cancel the common factor of $8$ .

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Step 5.3.8.1

Cancel the common factor.

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

Step 5.3.8.2

Rewrite the expression.

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

Step 5.4

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

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