Algebra Examples

$\log (\sqrt[3]{\frac{{(x + 5)}^{4}}{x^{2}}})$

Step 1

Rewrite $\sqrt[3]{\frac{{(x + 5)}^{4}}{x^{2}}}$ as $\frac{\sqrt[3]{{(x + 5)}^{4}}}{\sqrt[3]{x^{2}}}$ .

$\log (\frac{\sqrt[3]{{(x + 5)}^{4}}}{\sqrt[3]{x^{2}}})$

Step 2

Simplify the numerator.

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Step 2.1

Factor out ${(x + 5)}^{3}$ .

$\log (\frac{\sqrt[3]{{(x + 5)}^{3} (x + 5)}}{\sqrt[3]{x^{2}}})$

Step 2.2

Pull terms out from under the radical.

$\log (\frac{(x + 5) \sqrt[3]{x + 5}}{\sqrt[3]{x^{2}}})$

Step 2.3

Apply the distributive property.

$\log (\frac{x \sqrt[3]{x + 5} + 5 \sqrt[3]{x + 5}}{\sqrt[3]{x^{2}}})$

Step 2.4

Factor $\sqrt[3]{x + 5}$ out of $x \sqrt[3]{x + 5} + 5 \sqrt[3]{x + 5}$ .

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Step 2.4.1

Factor $\sqrt[3]{x + 5}$ out of $x \sqrt[3]{x + 5}$ .

$\log (\frac{\sqrt[3]{x + 5} x + 5 \sqrt[3]{x + 5}}{\sqrt[3]{x^{2}}})$

Step 2.4.2

Factor $\sqrt[3]{x + 5}$ out of $5 \sqrt[3]{x + 5}$ .

$\log (\frac{\sqrt[3]{x + 5} x + \sqrt[3]{x + 5} \cdot 5}{\sqrt[3]{x^{2}}})$

Step 2.4.3

Factor $\sqrt[3]{x + 5}$ out of $\sqrt[3]{x + 5} x + \sqrt[3]{x + 5} \cdot 5$ .

$\log (\frac{\sqrt[3]{x + 5} (x + 5)}{\sqrt[3]{x^{2}}})$

Step 3

Multiply $\frac{\sqrt[3]{x + 5} (x + 5)}{\sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$\log (\frac{\sqrt[3]{x + 5} (x + 5)}{\sqrt[3]{x^{2}}} \cdot \frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}})$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt[3]{x + 5} (x + 5)}{\sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{\sqrt[3]{x^{2}} {\sqrt[3]{x^{2}}}^{2}})$

Step 4.2

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

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{1} {\sqrt[3]{x^{2}}}^{2}})$

Step 4.3

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

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{1 + 2}})$

Step 4.4

Add $1$ and $2$ .

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{3}})$

Step 4.5

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

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Step 4.5.1

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

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{{(x^{\frac{2}{3}})}^{3}})$

Step 4.5.2

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

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2}{3} \cdot 3}})$

Step 4.5.3

Combine $\frac{2}{3}$ and $3$ .

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2 \cdot 3}{3}}})$

Step 4.5.4

Multiply $2$ by $3$ .

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{6}{3}}})$

Step 4.5.5

Cancel the common factor of $6$ and $3$ .

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Step 4.5.5.1

Factor $3$ out of $6$ .

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3}}})$

Step 4.5.5.2

Cancel the common factors.

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Step 4.5.5.2.1

Factor $3$ out of $3$ .

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3 (1)}}})$

Step 4.5.5.2.2

Cancel the common factor.

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{3 \cdot 2}{3 \cdot 1}}})$

Step 4.5.5.2.3

Rewrite the expression.

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{x^{\frac{2}{1}}})$

Step 4.5.5.2.4

Divide $2$ by $1$ .

$\log (\frac{\sqrt[3]{x + 5} (x + 5) {\sqrt[3]{x^{2}}}^{2}}{x^{2}})$

Step 5

Simplify the numerator.

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Step 5.1

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

$\log (\frac{\sqrt[3]{x + 5} (x + 5) \sqrt[3]{{(x^{2})}^{2}}}{x^{2}})$

Step 5.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 5.2.1

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

$\log (\frac{\sqrt[3]{x + 5} (x + 5) \sqrt[3]{x^{2 \cdot 2}}}{x^{2}})$

Step 5.2.2

Multiply $2$ by $2$ .

$\log (\frac{\sqrt[3]{x + 5} (x + 5) \sqrt[3]{x^{4}}}{x^{2}})$

Step 5.3

Factor out $x^{3}$ .

$\log (\frac{\sqrt[3]{x + 5} (x + 5) \sqrt[3]{x^{3} x}}{x^{2}})$

Step 5.4

Pull terms out from under the radical.

$\log (\frac{\sqrt[3]{x + 5} (x + 5) x \sqrt[3]{x}}{x^{2}})$

Step 5.5

Combine using the product rule for radicals.

$\log (\frac{(x + 5) x \sqrt[3]{(x + 5) x}}{x^{2}})$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $x$ and $x^{2}$ .

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Step 6.1.1

Factor $x$ out of $(x + 5) x \sqrt[3]{(x + 5) x}$ .

$\log (\frac{x ((x + 5) \sqrt[3]{(x + 5) x})}{x^{2}})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $x$ out of $x^{2}$ .

$\log (\frac{x ((x + 5) \sqrt[3]{(x + 5) x})}{x \cdot x})$

Step 6.1.2.2

Cancel the common factor.

$\log (\frac{x ((x + 5) \sqrt[3]{(x + 5) x})}{x \cdot x})$

Step 6.1.2.3

Rewrite the expression.

$\log (\frac{(x + 5) \sqrt[3]{(x + 5) x}}{x})$

Step 6.2

Reorder factors in $\log (\frac{(x + 5) \sqrt[3]{(x + 5) x}}{x})$ .

$\log (\frac{\sqrt[3]{x (x + 5)} (x + 5)}{x})$