Precalculus Examples

$\log_{5} (\frac{\sqrt[8]{m} \sqrt[3]{n}}{k^{2}})$

Step 1

Rewrite the expression using the least common index of $24$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{m}$ as $m^{\frac{1}{8}}$ .

$\log_{5} (\frac{m^{\frac{1}{8}} \sqrt[3]{n}}{k^{2}})$

Step 1.2

Rewrite $m^{\frac{1}{8}}$ as $m^{\frac{3}{24}}$ .

$\log_{5} (\frac{m^{\frac{3}{24}} \sqrt[3]{n}}{k^{2}})$

Step 1.3

Rewrite $m^{\frac{3}{24}}$ as $\sqrt[24]{m^{3}}$ .

$\log_{5} (\frac{\sqrt[24]{m^{3}} \sqrt[3]{n}}{k^{2}})$

Step 1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{n}$ as $n^{\frac{1}{3}}$ .

$\log_{5} (\frac{\sqrt[24]{m^{3}} n^{\frac{1}{3}}}{k^{2}})$

Step 1.5

Rewrite $n^{\frac{1}{3}}$ as $n^{\frac{8}{24}}$ .

$\log_{5} (\frac{\sqrt[24]{m^{3}} n^{\frac{8}{24}}}{k^{2}})$

Step 1.6

Rewrite $n^{\frac{8}{24}}$ as $\sqrt[24]{n^{8}}$ .

$\log_{5} (\frac{\sqrt[24]{m^{3}} \sqrt[24]{n^{8}}}{k^{2}})$

Step 2

Combine using the product rule for radicals.

$\log_{5} (\frac{\sqrt[24]{m^{3} n^{8}}}{k^{2}})$