Solve for a ( cube root of x^5)/((x^(1/3))^5)=x^a

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

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

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

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

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

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

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

Simplify the right side.

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

Simplify the numerator.

Factor out $x^{3}$ .

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

Pull terms out from under the radical.

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

Simplify the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $5$ .

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

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

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

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

Simplify the left side.

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

Cancel the common factor.

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

Divide $a$ by $1$ .

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