Algebra Examples

$\frac{\sqrt[6]{x^{5}}}{{(x^{\frac{1}{6}})}^{5}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = \frac{\sqrt[6]{x^{5}}}{{(x^{\frac{1}{6}})}^{5}}$ .

$x^{a} = \frac{\sqrt[6]{x^{5}}}{{(x^{\frac{1}{6}})}^{5}}$

Step 2

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

$\ln (x^{a}) = \ln (\frac{\sqrt[6]{x^{5}}}{{(x^{\frac{1}{6}})}^{5}})$

Step 3

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

$a \ln (x) = \ln (\frac{\sqrt[6]{x^{5}}}{{(x^{\frac{1}{6}})}^{5}})$

Step 4

Simplify the right side.

Tap for more steps...

Step 4.1

Multiply the exponents in ${(x^{\frac{1}{6}})}^{5}$ .

Tap for more steps...

Step 4.1.1

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

$a \ln (x) = \ln (\frac{\sqrt[6]{x^{5}}}{x^{\frac{1}{6} \cdot 5}})$

Step 4.1.2

Combine $\frac{1}{6}$ and $5$ .

$a \ln (x) = \ln (\frac{\sqrt[6]{x^{5}}}{x^{\frac{5}{6}}})$

Step 5

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[6]{x^{5}}}{x^{\frac{5}{6}}})$ by $\ln (x)$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $a \ln (x) = \ln (\frac{\sqrt[6]{x^{5}}}{x^{\frac{5}{6}}})$ by $\ln (x)$ .

$\frac{a \ln (x)}{\ln (x)} = \frac{\ln (\frac{\sqrt[6]{x^{5}}}{x^{\frac{5}{6}}})}{\ln (x)}$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

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

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

$\frac{a \ln (x)}{\ln (x)} = \frac{\ln (\frac{\sqrt[6]{x^{5}}}{x^{\frac{5}{6}}})}{\ln (x)}$

Step 5.2.1.2

Divide $a$ by $1$ .

$a = \frac{\ln (\frac{\sqrt[6]{x^{5}}}{x^{\frac{5}{6}}})}{\ln (x)}$