Precalculus Examples

$\log_{2} (2 \sqrt{8} \cdot (\sqrt[3]{\frac{1}{4}}))$

Step 1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 1.1

Factor $4$ out of $8$ .

$\log_{2} (2 \sqrt{4 (2)} \cdot \sqrt[3]{\frac{1}{4}})$

Step 1.2

Rewrite $4$ as $2^{2}$ .

$\log_{2} (2 \sqrt{2^{2} \cdot 2} \cdot \sqrt[3]{\frac{1}{4}})$

Step 2

Pull terms out from under the radical.

$\log_{2} (2 (2 \sqrt{2}) \cdot \sqrt[3]{\frac{1}{4}})$

Step 3

Multiply $2$ by $2$ .

$\log_{2} (4 \sqrt{2} \cdot \sqrt[3]{\frac{1}{4}})$

Step 4

Rewrite $\sqrt[3]{\frac{1}{4}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{4}}$ .

$\log_{2} (4 \sqrt{2} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{4}})$

Step 5

Any root of $1$ is $1$ .

$\log_{2} (4 \sqrt{2} \cdot \frac{1}{\sqrt[3]{4}})$

Step 6

Multiply $\frac{1}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

$\log_{2} (4 \sqrt{2} \cdot (\frac{1}{\sqrt[3]{4}} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}))$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{\sqrt[3]{4}}$ by $\frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{2}}$ .

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

Step 7.2

Raise $\sqrt[3]{4}$ to the power of $1$ .

$\log_{2} (4 \sqrt{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{2}})$

Step 7.3

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

$\log_{2} (4 \sqrt{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{{\sqrt[3]{4}}^{1 + 2}})$

Step 7.4

Add $1$ and $2$ .

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

Step 7.5

Rewrite ${\sqrt[3]{4}}^{3}$ as $4$ .

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Step 7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\log_{2} (4 \sqrt{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{{(4^{\frac{1}{3}})}^{3}})$

Step 7.5.2

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

$\log_{2} (4 \sqrt{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{4^{\frac{1}{3} \cdot 3}})$

Step 7.5.3

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

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

Step 7.5.4

Cancel the common factor of $3$ .

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Step 7.5.4.1

Cancel the common factor.

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

Step 7.5.4.2

Rewrite the expression.

$\log_{2} (4 \sqrt{2} \cdot \frac{{\sqrt[3]{4}}^{2}}{4^{1}})$

Step 7.5.5

Evaluate the exponent.

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

Step 8

Simplify the numerator.

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Step 8.1

Rewrite ${\sqrt[3]{4}}^{2}$ as $\sqrt[3]{4^{2}}$ .

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

Step 8.2

Raise $4$ to the power of $2$ .

$\log_{2} (4 \sqrt{2} \cdot \frac{\sqrt[3]{16}}{4})$

Step 8.3

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 8.3.1

Factor $8$ out of $16$ .

$\log_{2} (4 \sqrt{2} \cdot \frac{\sqrt[3]{8 (2)}}{4})$

Step 8.3.2

Rewrite $8$ as $2^{3}$ .

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

Step 8.4

Pull terms out from under the radical.

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

Step 9

Cancel the common factor of $4$ .

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Step 9.1

Factor $4$ out of $4 \sqrt{2}$ .

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

Step 9.2

Cancel the common factor.

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

Step 9.3

Rewrite the expression.

$\log_{2} (\sqrt{2} \cdot (2 \sqrt[3]{2}))$

Step 10

Rewrite the expression using the least common index of $6$ .

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Step 10.1

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

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

Step 10.2

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{3}{6}}$ .

$\log_{2} (2 (2^{\frac{3}{6}} \sqrt[3]{2}))$

Step 10.3

Rewrite $2^{\frac{3}{6}}$ as $\sqrt[6]{2^{3}}$ .

$\log_{2} (2 (\sqrt[6]{2^{3}} \sqrt[3]{2}))$

Step 10.4

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

$\log_{2} (2 (\sqrt[6]{2^{3}} \cdot 2^{\frac{1}{3}}))$

Step 10.5

Rewrite $2^{\frac{1}{3}}$ as $2^{\frac{2}{6}}$ .

$\log_{2} (2 (\sqrt[6]{2^{3}} \cdot 2^{\frac{2}{6}}))$

Step 10.6

Rewrite $2^{\frac{2}{6}}$ as $\sqrt[6]{2^{2}}$ .

$\log_{2} (2 (\sqrt[6]{2^{3}} \sqrt[6]{2^{2}}))$

Step 11

Combine using the product rule for radicals.

$\log_{2} (2 \sqrt[6]{2^{3} \cdot 2^{2}})$

Step 12

Multiply $2^{3}$ by $2^{2}$ by adding the exponents.

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Step 12.1

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

$\log_{2} (2 \sqrt[6]{2^{3 + 2}})$

Step 12.2

Add $3$ and $2$ .

$\log_{2} (2 \sqrt[6]{2^{5}})$

Step 13

Raise $2$ to the power of $5$ .

$\log_{2} (2 \sqrt[6]{32})$

Step 14

Rewrite $\log_{2} (2 \sqrt[6]{32})$ using the change of base formula.

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Step 14.1

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 14.2

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

$\frac{\log (2 \sqrt[6]{32})}{\log (2)}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (2 \sqrt[6]{32})}{\log (2)}$

Decimal Form:

$1.8\overline{3}$