Precalculus Examples

$\log (\sqrt[9]{\frac{x}{z}})$

Step 1

Rewrite $\sqrt[9]{\frac{x}{z}}$ as $\frac{\sqrt[9]{x}}{\sqrt[9]{z}}$ .

$\log (\frac{\sqrt[9]{x}}{\sqrt[9]{z}})$

Step 2

Multiply $\frac{\sqrt[9]{x}}{\sqrt[9]{z}}$ by $\frac{{\sqrt[9]{z}}^{8}}{{\sqrt[9]{z}}^{8}}$ .

$\log (\frac{\sqrt[9]{x}}{\sqrt[9]{z}} \cdot \frac{{\sqrt[9]{z}}^{8}}{{\sqrt[9]{z}}^{8}})$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[9]{x}}{\sqrt[9]{z}}$ by $\frac{{\sqrt[9]{z}}^{8}}{{\sqrt[9]{z}}^{8}}$ .

$\log (\frac{\sqrt[9]{x} {\sqrt[9]{z}}^{8}}{\sqrt[9]{z} {\sqrt[9]{z}}^{8}})$

Step 3.2

Raise $\sqrt[9]{z}$ to the power of $1$ .

$\log (\frac{\sqrt[9]{x} {\sqrt[9]{z}}^{8}}{{\sqrt[9]{z}}^{1} {\sqrt[9]{z}}^{8}})$

Step 3.3

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

$\log (\frac{\sqrt[9]{x} {\sqrt[9]{z}}^{8}}{{\sqrt[9]{z}}^{1 + 8}})$

Step 3.4

Add $1$ and $8$ .

$\log (\frac{\sqrt[9]{x} {\sqrt[9]{z}}^{8}}{{\sqrt[9]{z}}^{9}})$

Step 3.5

Rewrite ${\sqrt[9]{z}}^{9}$ as $z$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{z}$ as $z^{\frac{1}{9}}$ .

$\log (\frac{\sqrt[9]{x} {\sqrt[9]{z}}^{8}}{{(z^{\frac{1}{9}})}^{9}})$

Step 3.5.2

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

$\log (\frac{\sqrt[9]{x} {\sqrt[9]{z}}^{8}}{z^{\frac{1}{9} \cdot 9}})$

Step 3.5.3

Combine $\frac{1}{9}$ and $9$ .

$\log (\frac{\sqrt[9]{x} {\sqrt[9]{z}}^{8}}{z^{\frac{9}{9}}})$

Step 3.5.4

Cancel the common factor of $9$ .

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Step 3.5.4.1

Cancel the common factor.

$\log (\frac{\sqrt[9]{x} {\sqrt[9]{z}}^{8}}{z^{\frac{9}{9}}})$

Step 3.5.4.2

Rewrite the expression.

$\log (\frac{\sqrt[9]{x} {\sqrt[9]{z}}^{8}}{z^{1}})$

Step 3.5.5

Simplify.

$\log (\frac{\sqrt[9]{x} {\sqrt[9]{z}}^{8}}{z})$

Step 4

Rewrite ${\sqrt[9]{z}}^{8}$ as $\sqrt[9]{z^{8}}$ .

$\log (\frac{\sqrt[9]{x} \sqrt[9]{z^{8}}}{z})$

Step 5

Combine using the product rule for radicals.

$\log (\frac{\sqrt[9]{x z^{8}}}{z})$