Algebra Examples

$f (x) = {(\frac{x^{5}}{10})}^{\frac{1}{7}}$

Step 1

Write $f (x) = {(\frac{x^{5}}{10})}^{\frac{1}{7}}$ as an equation.

$y = {(\frac{x^{5}}{10})}^{\frac{1}{7}}$

Step 2

Interchange the variables.

$x = {(\frac{y^{5}}{10})}^{\frac{1}{7}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as ${(\frac{y^{5}}{10})}^{\frac{1}{7}} = x$ .

${(\frac{y^{5}}{10})}^{\frac{1}{7}} = x$

Step 3.2

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

${({(\frac{y^{5}}{10})}^{\frac{1}{7}})}^{7} = x^{7}$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify ${({(\frac{y^{5}}{10})}^{\frac{1}{7}})}^{7}$ .

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Step 3.3.1.1

Multiply the exponents in ${({(\frac{y^{5}}{10})}^{\frac{1}{7}})}^{7}$ .

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Step 3.3.1.1.1

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

${(\frac{y^{5}}{10})}^{\frac{1}{7} \cdot 7} = x^{7}$

Step 3.3.1.1.2

Cancel the common factor of $7$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

${(\frac{y^{5}}{10})}^{\frac{1}{7} \cdot 7} = x^{7}$

Step 3.3.1.1.2.2

Rewrite the expression.

${(\frac{y^{5}}{10})}^{1} = x^{7}$

Step 3.3.1.2

Simplify.

$\frac{y^{5}}{10} = x^{7}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Multiply both sides of the equation by $10$ .

$10 \frac{y^{5}}{10} = 10 x^{7}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $10$ .

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Step 3.4.2.1.1

Cancel the common factor.

$10 \frac{y^{5}}{10} = 10 x^{7}$

Step 3.4.2.1.2

Rewrite the expression.

$y^{5} = 10 x^{7}$

Step 3.4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[5]{10 x^{7}}$

Step 3.4.4

Simplify $\sqrt[5]{10 x^{7}}$ .

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Step 3.4.4.1

Rewrite $10 x^{7}$ as $x^{5} \cdot (10 x^{2})$ .

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Step 3.4.4.1.1

Factor out $x^{5}$ .

$y = \sqrt[5]{10 (x^{5} x^{2})}$

Step 3.4.4.1.2

Reorder $10$ and $x^{5}$ .

$y = \sqrt[5]{x^{5} \cdot 10 x^{2}}$

Step 3.4.4.1.3

Add parentheses.

$y = \sqrt[5]{x^{5} \cdot (10 x^{2})}$

Step 3.4.4.2

Pull terms out from under the radical.

$y = x \sqrt[5]{10 x^{2}}$

Step 4

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

$f^{- 1} (x) = x \sqrt[5]{10 x^{2}}$

Step 5

Verify if $f^{- 1} (x) = x \sqrt[5]{10 x^{2}}$ is the inverse of $f (x) = {(\frac{x^{5}}{10})}^{\frac{1}{7}}$ .

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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 $f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) \sqrt[5]{10 {({(\frac{x^{5}}{10})}^{\frac{1}{7}})}^{2}}$

Step 5.2.3

Apply basic rules of exponents.

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Step 5.2.3.1

Apply the product rule to $\frac{x^{5}}{10}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{{(x^{5})}^{\frac{1}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 {({(\frac{x^{5}}{10})}^{\frac{1}{7}})}^{2}}$

Step 5.2.3.2

Multiply the exponents in ${(x^{5})}^{\frac{1}{7}}$ .

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Step 5.2.3.2.1

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{5 (\frac{1}{7})}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 {({(\frac{x^{5}}{10})}^{\frac{1}{7}})}^{2}}$

Step 5.2.3.2.2

Combine $5$ and $\frac{1}{7}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 {({(\frac{x^{5}}{10})}^{\frac{1}{7}})}^{2}}$

Step 5.2.3.3

Multiply the exponents in ${({(\frac{x^{5}}{10})}^{\frac{1}{7}})}^{2}$ .

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Step 5.2.3.3.1

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 {(\frac{x^{5}}{10})}^{\frac{1}{7} \cdot 2}}$

Step 5.2.3.3.2

Combine $\frac{1}{7}$ and $2$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 {(\frac{x^{5}}{10})}^{\frac{2}{7}}}$

Step 5.2.3.4

Apply the product rule to $\frac{x^{5}}{10}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 (\frac{{(x^{5})}^{\frac{2}{7}}}{10^{\frac{2}{7}}})}$

Step 5.2.3.5

Multiply the exponents in ${(x^{5})}^{\frac{2}{7}}$ .

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Step 5.2.3.5.1

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 (\frac{x^{5 (\frac{2}{7})}}{10^{\frac{2}{7}}})}$

Step 5.2.3.5.2

Multiply $5 (\frac{2}{7})$ .

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Step 5.2.3.5.2.1

Combine $5$ and $\frac{2}{7}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 (\frac{x^{\frac{5 \cdot 2}{7}}}{10^{\frac{2}{7}}})}$

Step 5.2.3.5.2.2

Multiply $5$ by $2$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 (\frac{x^{\frac{10}{7}}}{10^{\frac{2}{7}}})}$

Step 5.2.4

Combine $10$ and $\frac{x^{\frac{10}{7}}}{10^{\frac{2}{7}}}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{\frac{10 x^{\frac{10}{7}}}{10^{\frac{2}{7}}}}$

Step 5.2.5

Reduce the expression by cancelling the common factors.

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Step 5.2.5.1

Move $10^{\frac{2}{7}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10 x^{\frac{10}{7}} \cdot 10^{- \frac{2}{7}}}$

Step 5.2.5.2

Multiply $10$ by $10^{- \frac{2}{7}}$ by adding the exponents.

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Step 5.2.5.2.1

Move $10^{- \frac{2}{7}}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10^{- \frac{2}{7}} \cdot (10 x^{\frac{10}{7}})}$

Step 5.2.5.2.2

Multiply $10^{- \frac{2}{7}}$ by $10$ .

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Step 5.2.5.2.2.1

Raise $10$ to the power of $1$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10^{- \frac{2}{7}} \cdot (10 x^{\frac{10}{7}})}$

Step 5.2.5.2.2.2

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10^{- \frac{2}{7} + 1} x^{\frac{10}{7}}}$

Step 5.2.5.2.3

Write $1$ as a fraction with a common denominator.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10^{- \frac{2}{7} + \frac{7}{7}} x^{\frac{10}{7}}}$

Step 5.2.5.2.4

Combine the numerators over the common denominator.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10^{\frac{- 2 + 7}{7}} x^{\frac{10}{7}}}$

Step 5.2.5.2.5

Add $- 2$ and $7$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{10^{\frac{5}{7}} x^{\frac{10}{7}}}$

Step 5.2.6

Rewrite $10^{\frac{5}{7}} x^{\frac{10}{7}}$ as ${(10^{\frac{1}{7}} x^{\frac{2}{7}})}^{5}$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot \sqrt[5]{{(10^{\frac{1}{7}} x^{\frac{2}{7}})}^{5}}$

Step 5.2.7

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = \frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}} \cdot (10^{\frac{1}{7}} x^{\frac{2}{7}})$

Step 5.2.8

Rewrite using the commutative property of multiplication.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = 10^{\frac{1}{7}} (\frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}}) \cdot x^{\frac{2}{7}}$

Step 5.2.9

Cancel the common factor of $10^{\frac{1}{7}}$ .

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Step 5.2.9.1

Cancel the common factor.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = 10^{\frac{1}{7}} (\frac{x^{\frac{5}{7}}}{10^{\frac{1}{7}}}) \cdot x^{\frac{2}{7}}$

Step 5.2.9.2

Rewrite the expression.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = x^{\frac{5}{7}} x^{\frac{2}{7}}$

Step 5.2.10

Multiply $x^{\frac{5}{7}}$ by $x^{\frac{2}{7}}$ by adding the exponents.

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Step 5.2.10.1

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

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = x^{\frac{5}{7} + \frac{2}{7}}$

Step 5.2.10.2

Combine the numerators over the common denominator.

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = x^{\frac{5 + 2}{7}}$

Step 5.2.10.3

Add $5$ and $2$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = x^{\frac{7}{7}}$

Step 5.2.10.4

Divide $7$ by $7$ .

$f^{- 1} ({(\frac{x^{5}}{10})}^{\frac{1}{7}}) = x$

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 $f (x \sqrt[5]{10 x^{2}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{{(x \sqrt[5]{10 x^{2}})}^{5}}{10})}^{\frac{1}{7}}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Apply the product rule to $x \sqrt[5]{10 x^{2}}$ .

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} {\sqrt[5]{10 x^{2}}}^{5}}{10})}^{\frac{1}{7}}$

Step 5.3.3.2

Rewrite ${\sqrt[5]{10 x^{2}}}^{5}$ as $10 x^{2}$ .

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Step 5.3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{10 x^{2}}$ as ${(10 x^{2})}^{\frac{1}{5}}$ .

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} {({(10 x^{2})}^{\frac{1}{5}})}^{5}}{10})}^{\frac{1}{7}}$

Step 5.3.3.2.2

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

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} {(10 x^{2})}^{\frac{1}{5} \cdot 5}}{10})}^{\frac{1}{7}}$

Step 5.3.3.2.3

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

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} {(10 x^{2})}^{\frac{5}{5}}}{10})}^{\frac{1}{7}}$

Step 5.3.3.2.4

Cancel the common factor of $5$ .

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Step 5.3.3.2.4.1

Cancel the common factor.

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} {(10 x^{2})}^{\frac{5}{5}}}{10})}^{\frac{1}{7}}$

Step 5.3.3.2.4.2

Rewrite the expression.

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} (10 x^{2})}{10})}^{\frac{1}{7}}$

Step 5.3.3.2.5

Simplify.

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} (10 x^{2})}{10})}^{\frac{1}{7}}$

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{5} \cdot (10 x^{2})}{10})}^{\frac{1}{7}}$

Step 5.3.3.3

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

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Step 5.3.3.3.1

Move $x^{2}$ .

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{2} x^{5} \cdot 10}{10})}^{\frac{1}{7}}$

Step 5.3.3.3.2

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

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{2 + 5} \cdot 10}{10})}^{\frac{1}{7}}$

Step 5.3.3.3.3

Add $2$ and $5$ .

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{7} \cdot 10}{10})}^{\frac{1}{7}}$

Step 5.3.4

Reduce the expression by cancelling the common factors.

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Step 5.3.4.1

Cancel the common factor of $10$ .

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Step 5.3.4.1.1

Cancel the common factor.

$f (x \sqrt[5]{10 x^{2}}) = {(\frac{x^{7} \cdot 10}{10})}^{\frac{1}{7}}$

Step 5.3.4.1.2

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

$f (x \sqrt[5]{10 x^{2}}) = {(x^{7})}^{\frac{1}{7}}$

Step 5.3.4.2

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

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Step 5.3.4.2.1

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

$f (x \sqrt[5]{10 x^{2}}) = x^{7 (\frac{1}{7})}$

Step 5.3.4.2.2

Cancel the common factor of $7$ .

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Step 5.3.4.2.2.1

Cancel the common factor.

$f (x \sqrt[5]{10 x^{2}}) = x^{7 (\frac{1}{7})}$

Step 5.3.4.2.2.2

Rewrite the expression.

$f (x \sqrt[5]{10 x^{2}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x \sqrt[5]{10 x^{2}}$ is the inverse of $f (x) = {(\frac{x^{5}}{10})}^{\frac{1}{7}}$ .

$f^{- 1} (x) = x \sqrt[5]{10 x^{2}}$