Precalculus Examples

$\log_{3} (\sqrt{\frac{r q}{6}})$

Rewrite $\sqrt{\frac{r q}{6}}$ as $\frac{\sqrt{r q}}{\sqrt{6}}$ .

$\log_{3} (\frac{\sqrt{r q}}{\sqrt{6}})$

Multiply $\frac{\sqrt{r q}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\log_{3} (\frac{\sqrt{r q}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Combine and simplify the denominator.

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Multiply $\frac{\sqrt{r q}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\log_{3} (\frac{\sqrt{r q} \sqrt{6}}{\sqrt{6} \sqrt{6}})$

Raise $\sqrt{6}$ to the power of $1$ .

$\log_{3} (\frac{\sqrt{r q} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}})$

Raise $\sqrt{6}$ to the power of $1$ .

$\log_{3} (\frac{\sqrt{r q} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{3} (\frac{\sqrt{r q} \sqrt{6}}{{\sqrt{6}}^{1 + 1}})$

Add $1$ and $1$ .

$\log_{3} (\frac{\sqrt{r q} \sqrt{6}}{{\sqrt{6}}^{2}})$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\log_{3} (\frac{\sqrt{r q} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}})$

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

$\log_{3} (\frac{\sqrt{r q} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}})$

Combine $\frac{1}{2}$ and $2$ .

$\log_{3} (\frac{\sqrt{r q} \sqrt{6}}{6^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\log_{3} (\frac{\sqrt{r q} \sqrt{6}}{6^{\frac{2}{2}}})$

Rewrite the expression.

$\log_{3} (\frac{\sqrt{r q} \sqrt{6}}{6^{1}})$

Evaluate the exponent.

$\log_{3} (\frac{\sqrt{r q} \sqrt{6}}{6})$

Combine using the product rule for radicals.

$\log_{3} (\frac{\sqrt{r q \cdot 6}}{6})$

Reorder factors in $\log_{3} (\frac{\sqrt{r q \cdot 6}}{6})$ .

$\log_{3} (\frac{\sqrt{6 r q}}{6})$