Precalculus Examples

$\log_{7} (\frac{\sqrt[6]{p} \sqrt[5]{q}}{t^{2}})$

Step 1

Rewrite the expression using the least common index of $30$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{p}$ as $p^{\frac{1}{6}}$ .

$\log_{7} (\frac{p^{\frac{1}{6}} \sqrt[5]{q}}{t^{2}})$

Step 1.2

Rewrite $p^{\frac{1}{6}}$ as $p^{\frac{5}{30}}$ .

$\log_{7} (\frac{p^{\frac{5}{30}} \sqrt[5]{q}}{t^{2}})$

Step 1.3

Rewrite $p^{\frac{5}{30}}$ as $\sqrt[30]{p^{5}}$ .

$\log_{7} (\frac{\sqrt[30]{p^{5}} \sqrt[5]{q}}{t^{2}})$

Step 1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{q}$ as $q^{\frac{1}{5}}$ .

$\log_{7} (\frac{\sqrt[30]{p^{5}} q^{\frac{1}{5}}}{t^{2}})$

Step 1.5

Rewrite $q^{\frac{1}{5}}$ as $q^{\frac{6}{30}}$ .

$\log_{7} (\frac{\sqrt[30]{p^{5}} q^{\frac{6}{30}}}{t^{2}})$

Step 1.6

Rewrite $q^{\frac{6}{30}}$ as $\sqrt[30]{q^{6}}$ .

$\log_{7} (\frac{\sqrt[30]{p^{5}} \sqrt[30]{q^{6}}}{t^{2}})$

Step 2

Combine using the product rule for radicals.

$\log_{7} (\frac{\sqrt[30]{p^{5} q^{6}}}{t^{2}})$