Algebra Examples

$\frac{\sqrt[3]{x^{5}}}{{(x^{5})}^{\frac{1}{3}}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{\sqrt[3]{x^{5}}}{{(x^{5})}^{\frac{1}{3}}}$ .

$x^{a} = \frac{\sqrt[3]{x^{5}}}{{(x^{5})}^{\frac{1}{3}}}$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (x^{a}) = \ln (\frac{\sqrt[3]{x^{5}}}{{(x^{5})}^{\frac{1}{3}}})$

Step 3

Expand $\ln (x^{a})$ by moving $a$ outside the logarithm.

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{5}}}{{(x^{5})}^{\frac{1}{3}}})$

Step 4

Simplify the right side.

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Step 4.1

Simplify $\ln (\frac{\sqrt[3]{x^{5}}}{{(x^{5})}^{\frac{1}{3}}})$ .

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Step 4.1.1

Simplify the numerator.

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Step 4.1.1.1

Factor out $x^{3}$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{3} x^{2}}}{{(x^{5})}^{\frac{1}{3}}})$

Step 4.1.1.2

Pull terms out from under the radical.

$a \ln (x) = \ln (\frac{x \sqrt[3]{x^{2}}}{{(x^{5})}^{\frac{1}{3}}})$

Step 4.1.2

Simplify the expression.

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Step 4.1.2.1

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

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Step 4.1.2.1.1

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

$a \ln (x) = \ln (\frac{x \sqrt[3]{x^{2}}}{x^{5 (\frac{1}{3})}})$

Step 4.1.2.1.2

Combine $5$ and $\frac{1}{3}$ .

$a \ln (x) = \ln (\frac{x \sqrt[3]{x^{2}}}{x^{\frac{5}{3}}})$

Step 4.1.2.2

Move $x^{1}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{5}{3}} x^{- 1}})$

Step 4.1.3

Multiply $x^{\frac{5}{3}}$ by $x^{- 1}$ by adding the exponents.

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Step 4.1.3.1

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

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{5}{3} - 1}})$

Step 4.1.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{5}{3} - 1 \cdot \frac{3}{3}}})$

Step 4.1.3.3

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

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{5}{3} + \frac{- 1 \cdot 3}{3}}})$

Step 4.1.3.4

Combine the numerators over the common denominator.

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{5 - 1 \cdot 3}{3}}})$

Step 4.1.3.5

Simplify the numerator.

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Step 4.1.3.5.1

Multiply $- 1$ by $3$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{5 - 3}{3}}})$

Step 4.1.3.5.2

Subtract $3$ from $5$ .

$a \ln (x) = \ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{2}{3}}})$

Step 5

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{2}{3}}})$ by $\ln (x)$ and simplify.

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Step 5.1

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{2}{3}}})$ by $\ln (x)$ .

$\frac{a \ln (x)}{\ln (x)} = \frac{\ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{2}{3}}})}{\ln (x)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $\ln (x)$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{a \ln (x)}{\ln (x)} = \frac{\ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{2}{3}}})}{\ln (x)}$

Step 5.2.1.2

Divide $a$ by $1$ .

$a = \frac{\ln (\frac{\sqrt[3]{x^{2}}}{x^{\frac{2}{3}}})}{\ln (x)}$