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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[3]{x}}{x^{\frac{1}{3}}})$ by $\ln (x)$ and simplify.

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[3]{x}}{x^{\frac{1}{3}}})$ by $\ln (x)$ .

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

Simplify the left side.

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

Cancel the common factor.

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

Divide $a$ by $1$ .

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