Algebra Examples

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

Step 1

Rewrite the equation as $x^{a} = \frac{\sqrt[5]{x^{2}}}{{(x^{2})}^{\frac{1}{5}}}$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify the right side.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Combine $2$ and $\frac{1}{5}$ .

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

Step 5

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[5]{x^{2}}}{x^{\frac{2}{5}}})$ by $\ln (x)$ and simplify.

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Step 5.1

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[5]{x^{2}}}{x^{\frac{2}{5}}})$ by $\ln (x)$ .

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

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.

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

Step 5.2.1.2

Divide $a$ by $1$ .

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