Precalculus Examples

$\log_{2} (\sqrt[3]{\frac{x^{2}}{y \sqrt{2}}})$

Step 1

Multiply $\frac{x^{2}}{y \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\log_{2} (\sqrt[3]{\frac{x^{2}}{y \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}})$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{x^{2}}{y \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{y \sqrt{2} \sqrt{2}}})$

Step 2.2

Move $\sqrt{2}$ .

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{y (\sqrt{2} \sqrt{2})}})$

Step 2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{y ({\sqrt{2}}^{1} \sqrt{2})}})$

Step 2.4

Raise $\sqrt{2}$ to the power of $1$ .

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{y ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}})$

Step 2.5

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

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

Step 2.6

Add $1$ and $1$ .

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{y {\sqrt{2}}^{2}}})$

Step 2.7

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

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Step 2.7.1

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

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{y {(2^{\frac{1}{2}})}^{2}}})$

Step 2.7.2

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

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{y \cdot 2^{\frac{1}{2} \cdot 2}}})$

Step 2.7.3

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

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{y \cdot 2^{\frac{2}{2}}}})$

Step 2.7.4

Cancel the common factor of $2$ .

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Step 2.7.4.1

Cancel the common factor.

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{y \cdot 2^{\frac{2}{2}}}})$

Step 2.7.4.2

Rewrite the expression.

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{y \cdot 2^{1}}})$

Step 2.7.5

Evaluate the exponent.

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{y \cdot 2}})$

Step 3

Move $2$ to the left of $y$ .

$\log_{2} (\sqrt[3]{\frac{x^{2} \sqrt{2}}{2 \cdot y}})$

Step 4

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

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}}}{\sqrt[3]{2 y}})$

Step 5

Multiply $\frac{\sqrt[3]{x^{2} \sqrt{2}}}{\sqrt[3]{2 y}}$ by $\frac{{\sqrt[3]{2 y}}^{2}}{{\sqrt[3]{2 y}}^{2}}$ .

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}}}{\sqrt[3]{2 y}} \cdot \frac{{\sqrt[3]{2 y}}^{2}}{{\sqrt[3]{2 y}}^{2}})$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt[3]{x^{2} \sqrt{2}}}{\sqrt[3]{2 y}}$ by $\frac{{\sqrt[3]{2 y}}^{2}}{{\sqrt[3]{2 y}}^{2}}$ .

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} {\sqrt[3]{2 y}}^{2}}{\sqrt[3]{2 y} {\sqrt[3]{2 y}}^{2}})$

Step 6.2

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

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} {\sqrt[3]{2 y}}^{2}}{{\sqrt[3]{2 y}}^{1} {\sqrt[3]{2 y}}^{2}})$

Step 6.3

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

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

Step 6.4

Add $1$ and $2$ .

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} {\sqrt[3]{2 y}}^{2}}{{\sqrt[3]{2 y}}^{3}})$

Step 6.5

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

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Step 6.5.1

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

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} {\sqrt[3]{2 y}}^{2}}{{({(2 y)}^{\frac{1}{3}})}^{3}})$

Step 6.5.2

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

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} {\sqrt[3]{2 y}}^{2}}{{(2 y)}^{\frac{1}{3} \cdot 3}})$

Step 6.5.3

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

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} {\sqrt[3]{2 y}}^{2}}{{(2 y)}^{\frac{3}{3}}})$

Step 6.5.4

Cancel the common factor of $3$ .

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Step 6.5.4.1

Cancel the common factor.

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} {\sqrt[3]{2 y}}^{2}}{{(2 y)}^{\frac{3}{3}}})$

Step 6.5.4.2

Rewrite the expression.

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} {\sqrt[3]{2 y}}^{2}}{{(2 y)}^{1}})$

Step 6.5.5

Simplify.

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} {\sqrt[3]{2 y}}^{2}}{2 y})$

Step 7

Simplify the numerator.

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Step 7.1

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

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} \sqrt[3]{{(2 y)}^{2}}}{2 y})$

Step 7.2

Apply the product rule to $2 y$ .

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} \sqrt[3]{2^{2} y^{2}}}{2 y})$

Step 7.3

Raise $2$ to the power of $2$ .

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2}} \sqrt[3]{4 y^{2}}}{2 y})$

Step 8

Combine using the product rule for radicals.

$\log_{2} (\frac{\sqrt[3]{x^{2} \sqrt{2} \cdot 4 y^{2}}}{2 y})$