Precalculus Examples

$\log (5) (\frac{\sqrt[7]{m} \sqrt[6]{n}}{k^{2}})$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the least common index of $42$ .

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{m}$ as $m^{\frac{1}{7}}$ .

$\log (5) \frac{m^{\frac{1}{7}} \sqrt[6]{n}}{k^{2}}$

Step 1.1.2

Rewrite $m^{\frac{1}{7}}$ as $m^{\frac{6}{42}}$ .

$\log (5) \frac{m^{\frac{6}{42}} \sqrt[6]{n}}{k^{2}}$

Step 1.1.3

Rewrite $m^{\frac{6}{42}}$ as $\sqrt[42]{m^{6}}$ .

$\log (5) \frac{\sqrt[42]{m^{6}} \sqrt[6]{n}}{k^{2}}$

Step 1.1.4

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

$\log (5) \frac{\sqrt[42]{m^{6}} n^{\frac{1}{6}}}{k^{2}}$

Step 1.1.5

Rewrite $n^{\frac{1}{6}}$ as $n^{\frac{7}{42}}$ .

$\log (5) \frac{\sqrt[42]{m^{6}} n^{\frac{7}{42}}}{k^{2}}$

Step 1.1.6

Rewrite $n^{\frac{7}{42}}$ as $\sqrt[42]{n^{7}}$ .

$\log (5) \frac{\sqrt[42]{m^{6}} \sqrt[42]{n^{7}}}{k^{2}}$

Step 1.2

Combine using the product rule for radicals.

$\log (5) \frac{\sqrt[42]{m^{6} n^{7}}}{k^{2}}$

Step 2

Combine $\log (5)$ and $\frac{\sqrt[42]{m^{6} n^{7}}}{k^{2}}$ .

$\frac{\log (5) \sqrt[42]{m^{6} n^{7}}}{k^{2}}$