Algebra Examples

$\log_{w} (\frac{{(x^{2} - 6)}^{4}}{\sqrt[3]{x^{2} + 8}})$

Step 1

Multiply $\frac{{(x^{2} - 6)}^{4}}{\sqrt[3]{x^{2} + 8}}$ by $\frac{{\sqrt[3]{x^{2} + 8}}^{2}}{{\sqrt[3]{x^{2} + 8}}^{2}}$ .

$\log_{w} (\frac{{(x^{2} - 6)}^{4}}{\sqrt[3]{x^{2} + 8}} \cdot \frac{{\sqrt[3]{x^{2} + 8}}^{2}}{{\sqrt[3]{x^{2} + 8}}^{2}})$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{{(x^{2} - 6)}^{4}}{\sqrt[3]{x^{2} + 8}}$ by $\frac{{\sqrt[3]{x^{2} + 8}}^{2}}{{\sqrt[3]{x^{2} + 8}}^{2}}$ .

$\log_{w} (\frac{{(x^{2} - 6)}^{4} {\sqrt[3]{x^{2} + 8}}^{2}}{\sqrt[3]{x^{2} + 8} {\sqrt[3]{x^{2} + 8}}^{2}})$

Step 2.2

Raise $\sqrt[3]{x^{2} + 8}$ to the power of $1$ .

$\log_{w} (\frac{{(x^{2} - 6)}^{4} {\sqrt[3]{x^{2} + 8}}^{2}}{{\sqrt[3]{x^{2} + 8}}^{1} {\sqrt[3]{x^{2} + 8}}^{2}})$

Step 2.3

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

$\log_{w} (\frac{{(x^{2} - 6)}^{4} {\sqrt[3]{x^{2} + 8}}^{2}}{{\sqrt[3]{x^{2} + 8}}^{1 + 2}})$

Step 2.4

Add $1$ and $2$ .

$\log_{w} (\frac{{(x^{2} - 6)}^{4} {\sqrt[3]{x^{2} + 8}}^{2}}{{\sqrt[3]{x^{2} + 8}}^{3}})$

Step 2.5

Rewrite ${\sqrt[3]{x^{2} + 8}}^{3}$ as $x^{2} + 8$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2} + 8}$ as ${(x^{2} + 8)}^{\frac{1}{3}}$ .

$\log_{w} (\frac{{(x^{2} - 6)}^{4} {\sqrt[3]{x^{2} + 8}}^{2}}{{({(x^{2} + 8)}^{\frac{1}{3}})}^{3}})$

Step 2.5.2

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

$\log_{w} (\frac{{(x^{2} - 6)}^{4} {\sqrt[3]{x^{2} + 8}}^{2}}{{(x^{2} + 8)}^{\frac{1}{3} \cdot 3}})$

Step 2.5.3

Combine $\frac{1}{3}$ and $3$ .

$\log_{w} (\frac{{(x^{2} - 6)}^{4} {\sqrt[3]{x^{2} + 8}}^{2}}{{(x^{2} + 8)}^{\frac{3}{3}}})$

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

$\log_{w} (\frac{{(x^{2} - 6)}^{4} {\sqrt[3]{x^{2} + 8}}^{2}}{{(x^{2} + 8)}^{\frac{3}{3}}})$

Step 2.5.4.2

Rewrite the expression.

$\log_{w} (\frac{{(x^{2} - 6)}^{4} {\sqrt[3]{x^{2} + 8}}^{2}}{{(x^{2} + 8)}^{1}})$

Step 2.5.5

Simplify.

$\log_{w} (\frac{{(x^{2} - 6)}^{4} {\sqrt[3]{x^{2} + 8}}^{2}}{x^{2} + 8})$

Step 3

Rewrite ${\sqrt[3]{x^{2} + 8}}^{2}$ as $\sqrt[3]{{(x^{2} + 8)}^{2}}$ .

$\log_{w} (\frac{{(x^{2} - 6)}^{4} \sqrt[3]{{(x^{2} + 8)}^{2}}}{x^{2} + 8})$

Step 4

Reorder factors in $\log_{w} (\frac{{(x^{2} - 6)}^{4} \sqrt[3]{{(x^{2} + 8)}^{2}}}{x^{2} + 8})$ .

$\log_{w} (\frac{\sqrt[3]{{(x^{2} + 8)}^{2}} {(x^{2} - 6)}^{4}}{x^{2} + 8})$

Step 5

Rewrite $\log_{w} (\frac{\sqrt[3]{{(x^{2} + 8)}^{2}} {(x^{2} - 6)}^{4}}{x^{2} + 8})$ using the change of base formula.

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Step 5.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 5.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (\frac{\sqrt[3]{{(x^{2} + 8)}^{2}} {(x^{2} - 6)}^{4}}{x^{2} + 8})}{\log (w)}$