Solve the Rational Equation for a ( fifth root of x^3)/((x^3)^(1/5))=x^a

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

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

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

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

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

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

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

Simplify the right side.

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

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

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

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

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

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[5]{x^{3}}}{x^{\frac{3}{5}}})$ by $\ln (x)$ and simplify.

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[5]{x^{3}}}{x^{\frac{3}{5}}})$ by $\ln (x)$ .

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

Simplify the left side.

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

Cancel the common factor.

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

Divide $a$ by $1$ .

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