Precalculus Examples

$\ln (x^{6 \sqrt[9]{y^{6}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{y^{6}}$ as $y^{\frac{6}{9}}$ .

$\ln (x^{6 y^{\frac{6}{9}}})$

Step 2

Cancel the common factor of $6$ and $9$ .

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Step 2.1

Factor $3$ out of $6$ .

$\ln (x^{6 y^{\frac{3 (2)}{9}}})$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $3$ out of $9$ .

$\ln (x^{6 y^{\frac{3 \cdot 2}{3 \cdot 3}}})$

Step 2.2.2

Cancel the common factor.

$\ln (x^{6 y^{\frac{3 \cdot 2}{3 \cdot 3}}})$

Step 2.2.3

Rewrite the expression.

$\ln (x^{6 y^{\frac{2}{3}}})$

Step 3

Expand $\ln (x^{6 y^{\frac{2}{3}}})$ by moving $6 y^{\frac{2}{3}}$ outside the logarithm.

$6 y^{\frac{2}{3}} \ln (x)$