Algebra Examples

$7 {(x - 3)}^{\frac{1}{5}}$

Step 1

Interchange the variables.

$x = 7 {(y - 3)}^{\frac{1}{5}}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $7 {(y - 3)}^{\frac{1}{5}} = x$ .

$7 {(y - 3)}^{\frac{1}{5}} = x$

Step 2.2

Divide each term in $7 {(y - 3)}^{\frac{1}{5}} = x$ by $7$ and simplify.

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Step 2.2.1

Divide each term in $7 {(y - 3)}^{\frac{1}{5}} = x$ by $7$ .

$\frac{7 {(y - 3)}^{\frac{1}{5}}}{7} = \frac{x}{7}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor.

$\frac{7 {(y - 3)}^{\frac{1}{5}}}{7} = \frac{x}{7}$

Step 2.2.2.2

Divide ${(y - 3)}^{\frac{1}{5}}$ by $1$ .

${(y - 3)}^{\frac{1}{5}} = \frac{x}{7}$

Step 2.3

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

${({(y - 3)}^{\frac{1}{5}})}^{5} = {(\frac{x}{7})}^{5}$

Step 2.4

Simplify the exponent.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify ${({(y - 3)}^{\frac{1}{5}})}^{5}$ .

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Step 2.4.1.1.1

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

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Step 2.4.1.1.1.1

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

${(y - 3)}^{\frac{1}{5} \cdot 5} = {(\frac{x}{7})}^{5}$

Step 2.4.1.1.1.2

Cancel the common factor of $5$ .

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Step 2.4.1.1.1.2.1

Cancel the common factor.

${(y - 3)}^{\frac{1}{5} \cdot 5} = {(\frac{x}{7})}^{5}$

Step 2.4.1.1.1.2.2

Rewrite the expression.

${(y - 3)}^{1} = {(\frac{x}{7})}^{5}$

Step 2.4.1.1.2

Simplify.

$y - 3 = {(\frac{x}{7})}^{5}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify ${(\frac{x}{7})}^{5}$ .

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Step 2.4.2.1.1

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

$y - 3 = \frac{x^{5}}{7^{5}}$

Step 2.4.2.1.2

Raise $7$ to the power of $5$ .

$y - 3 = \frac{x^{5}}{16807}$

Step 2.5

Add $3$ to both sides of the equation.

$y = \frac{x^{5}}{16807} + 3$

Step 3

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

$f^{- 1} (x) = \frac{x^{5}}{16807} + 3$

Step 4

Verify if $f^{- 1} (x) = \frac{x^{5}}{16807} + 3$ is the inverse of $f (x) = 7 {(x - 3)}^{\frac{1}{5}}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (7 {(x - 3)}^{\frac{1}{5}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (7 {(x - 3)}^{\frac{1}{5}}) = \frac{{(7 {(x - 3)}^{\frac{1}{5}})}^{5}}{16807} + 3$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

Apply the product rule to $7 {(x - 3)}^{\frac{1}{5}}$ .

$f^{- 1} (7 {(x - 3)}^{\frac{1}{5}}) = \frac{7^{5} {({(x - 3)}^{\frac{1}{5}})}^{5}}{16807} + 3$

Step 4.2.3.1.2

Raise $7$ to the power of $5$ .

$f^{- 1} (7 {(x - 3)}^{\frac{1}{5}}) = \frac{16807 {({(x - 3)}^{\frac{1}{5}})}^{5}}{16807} + 3$

Step 4.2.3.1.3

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

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Step 4.2.3.1.3.1

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

$f^{- 1} (7 {(x - 3)}^{\frac{1}{5}}) = \frac{16807 {(x - 3)}^{\frac{1}{5} \cdot 5}}{16807} + 3$

Step 4.2.3.1.3.2

Cancel the common factor of $5$ .

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Step 4.2.3.1.3.2.1

Cancel the common factor.

$f^{- 1} (7 {(x - 3)}^{\frac{1}{5}}) = \frac{16807 {(x - 3)}^{\frac{1}{5} \cdot 5}}{16807} + 3$

Step 4.2.3.1.3.2.2

Rewrite the expression.

$f^{- 1} (7 {(x - 3)}^{\frac{1}{5}}) = \frac{16807 (x - 3)}{16807} + 3$

Step 4.2.3.1.4

Simplify.

$f^{- 1} (7 {(x - 3)}^{\frac{1}{5}}) = \frac{16807 (x - 3)}{16807} + 3$

Step 4.2.3.2

Cancel the common factor of $16807$ .

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Step 4.2.3.2.1

Cancel the common factor.

$f^{- 1} (7 {(x - 3)}^{\frac{1}{5}}) = \frac{16807 (x - 3)}{16807} + 3$

Step 4.2.3.2.2

Divide $x - 3$ by $1$ .

$f^{- 1} (7 {(x - 3)}^{\frac{1}{5}}) = x - 3 + 3$

Step 4.2.4

Combine the opposite terms in $x - 3 + 3$ .

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Step 4.2.4.1

Add $- 3$ and $3$ .

$f^{- 1} (7 {(x - 3)}^{\frac{1}{5}}) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

$f^{- 1} (7 {(x - 3)}^{\frac{1}{5}}) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{x^{5}}{16807} + 3)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{5}}{16807} + 3) = 7 {((\frac{x^{5}}{16807} + 3) - 3)}^{\frac{1}{5}}$

Step 4.3.3

Simplify by adding terms.

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Step 4.3.3.1

Combine the opposite terms in $(\frac{x^{5}}{16807} + 3) - 3$ .

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Step 4.3.3.1.1

Subtract $3$ from $3$ .

$f (\frac{x^{5}}{16807} + 3) = 7 {(\frac{x^{5}}{16807} + 0)}^{\frac{1}{5}}$

Step 4.3.3.1.2

Add $\frac{x^{5}}{16807}$ and $0$ .

$f (\frac{x^{5}}{16807} + 3) = 7 {(\frac{x^{5}}{16807})}^{\frac{1}{5}}$

Step 4.3.3.2

Apply the product rule to $\frac{x^{5}}{16807}$ .

$f (\frac{x^{5}}{16807} + 3) = 7 (\frac{{(x^{5})}^{\frac{1}{5}}}{16807^{\frac{1}{5}}})$

Step 4.3.4

Simplify the numerator.

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Step 4.3.4.1

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

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Step 4.3.4.1.1

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

$f (\frac{x^{5}}{16807} + 3) = 7 (\frac{x^{5 (\frac{1}{5})}}{16807^{\frac{1}{5}}})$

Step 4.3.4.1.2

Cancel the common factor of $5$ .

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Step 4.3.4.1.2.1

Cancel the common factor.

$f (\frac{x^{5}}{16807} + 3) = 7 (\frac{x^{5 (\frac{1}{5})}}{16807^{\frac{1}{5}}})$

Step 4.3.4.1.2.2

Rewrite the expression.

$f (\frac{x^{5}}{16807} + 3) = 7 (\frac{x}{16807^{\frac{1}{5}}})$

Step 4.3.4.2

Simplify.

$f (\frac{x^{5}}{16807} + 3) = 7 (\frac{x}{16807^{\frac{1}{5}}})$

Step 4.3.5

Simplify the denominator.

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Step 4.3.5.1

Rewrite $16807$ as $7^{5}$ .

$f (\frac{x^{5}}{16807} + 3) = 7 (\frac{x}{{(7^{5})}^{\frac{1}{5}}})$

Step 4.3.5.2

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

$f (\frac{x^{5}}{16807} + 3) = 7 (\frac{x}{7^{5 (\frac{1}{5})}})$

Step 4.3.5.3

Cancel the common factor of $5$ .

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Step 4.3.5.3.1

Cancel the common factor.

$f (\frac{x^{5}}{16807} + 3) = 7 (\frac{x}{7^{5 (\frac{1}{5})}})$

Step 4.3.5.3.2

Rewrite the expression.

$f (\frac{x^{5}}{16807} + 3) = 7 (\frac{x}{7})$

Step 4.3.5.4

Evaluate the exponent.

$f (\frac{x^{5}}{16807} + 3) = 7 (\frac{x}{7})$

Step 4.3.6

Cancel the common factor of $7$ .

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Step 4.3.6.1

Cancel the common factor.

$f (\frac{x^{5}}{16807} + 3) = 7 (\frac{x}{7})$

Step 4.3.6.2

Rewrite the expression.

$f (\frac{x^{5}}{16807} + 3) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{5}}{16807} + 3$ is the inverse of $f (x) = 7 {(x - 3)}^{\frac{1}{5}}$ .

$f^{- 1} (x) = \frac{x^{5}}{16807} + 3$