Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log_{2} (\sqrt[4]{2})$ by moving $3$ inside the logarithm.

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

Step 1.2

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

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

Step 1.3

Raise $2$ to the power of $3$ .

${(1 + \log_{2} (\sqrt[4]{8}))}^{2}$

Step 2

Rewrite $\log_{2} (\sqrt[4]{8})$ using the change of base formula.

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Step 2.1

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 $0$ .

${(1 + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)})}^{2}$

Step 2.2

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

${(1 + \frac{\log (\sqrt[4]{8})}{\log (2)})}^{2}$

Step 3

Rewrite ${(1 + \frac{\log (\sqrt[4]{8})}{\log (2)})}^{2}$ as $(1 + \frac{\log (\sqrt[4]{8})}{\log (2)}) (1 + \frac{\log (\sqrt[4]{8})}{\log (2)})$ .

$(1 + \frac{\log (\sqrt[4]{8})}{\log (2)}) (1 + \frac{\log (\sqrt[4]{8})}{\log (2)})$

Step 4

Expand $(1 + \frac{\log (\sqrt[4]{8})}{\log (2)}) (1 + \frac{\log (\sqrt[4]{8})}{\log (2)})$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$1 (1 + \frac{\log (\sqrt[4]{8})}{\log (2)}) + \frac{\log (\sqrt[4]{8})}{\log (2)} (1 + \frac{\log (\sqrt[4]{8})}{\log (2)})$

Step 4.2

Apply the distributive property.

$1 \cdot 1 + 1 \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} (1 + \frac{\log (\sqrt[4]{8})}{\log (2)})$

Step 4.3

Apply the distributive property.

$1 \cdot 1 + 1 \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} \cdot 1 + \frac{\log (\sqrt[4]{8})}{\log (2)} \cdot \frac{\log (\sqrt[4]{8})}{\log (2)}$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $1$ by $1$ .

$1 + 1 \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} \cdot 1 + \frac{\log (\sqrt[4]{8})}{\log (2)} \cdot \frac{\log (\sqrt[4]{8})}{\log (2)}$

Step 5.1.2

Multiply $\frac{\log (\sqrt[4]{8})}{\log (2)}$ by $1$ .

$1 + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} \cdot 1 + \frac{\log (\sqrt[4]{8})}{\log (2)} \cdot \frac{\log (\sqrt[4]{8})}{\log (2)}$

Step 5.1.3

Multiply $\frac{\log (\sqrt[4]{8})}{\log (2)}$ by $1$ .

$1 + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} \cdot \frac{\log (\sqrt[4]{8})}{\log (2)}$

Step 5.1.4

Multiply $\frac{\log (\sqrt[4]{8})}{\log (2)} \cdot \frac{\log (\sqrt[4]{8})}{\log (2)}$ .

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Step 5.1.4.1

Multiply $\frac{\log (\sqrt[4]{8})}{\log (2)}$ by $\frac{\log (\sqrt[4]{8})}{\log (2)}$ .

$1 + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8}) \log (\sqrt[4]{8})}{\log (2) \log (2)}$

Step 5.1.4.2

Raise $\log (\sqrt[4]{8})$ to the power of $1$ .

$1 + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log^{1} (\sqrt[4]{8}) \log (\sqrt[4]{8})}{\log (2) \log (2)}$

Step 5.1.4.3

Raise $\log (\sqrt[4]{8})$ to the power of $1$ .

$1 + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log^{1} (\sqrt[4]{8}) \log^{1} (\sqrt[4]{8})}{\log (2) \log (2)}$

Step 5.1.4.4

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

$1 + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{{\log (\sqrt[4]{8})}^{1 + 1}}{\log (2) \log (2)}$

Step 5.1.4.5

Add $1$ and $1$ .

$1 + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log (2) \log (2)}$

Step 5.1.4.6

Raise $\log (2)$ to the power of $1$ .

$1 + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{1} (2) \log (2)}$

Step 5.1.4.7

Raise $\log (2)$ to the power of $1$ .

$1 + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{1} (2) \log^{1} (2)}$

Step 5.1.4.8

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

$1 + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{{\log (2)}^{1 + 1}}$

Step 5.1.4.9

Add $1$ and $1$ .

$1 + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log (\sqrt[4]{8})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Step 5.2

Combine the numerators over the common denominator.

$1 + \frac{\log (\sqrt[4]{8}) + \log (\sqrt[4]{8})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Step 6

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$1 + \frac{\log (\sqrt[4]{8} \sqrt[4]{8})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$1 + \frac{\log (\sqrt[4]{8 \cdot 8})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Step 7.2

Multiply $8$ by $8$ .

$1 + \frac{\log (\sqrt[4]{64})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Step 7.3

Rewrite $64$ as $2^{4} \cdot 4$ .

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Step 7.3.1

Factor $16$ out of $64$ .

$1 + \frac{\log (\sqrt[4]{16 (4)})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Step 7.3.2

Rewrite $16$ as $2^{4}$ .

$1 + \frac{\log (\sqrt[4]{2^{4} \cdot 4})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Step 7.4

Pull terms out from under the radical.

$1 + \frac{\log (2 \sqrt[4]{4})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Step 7.5

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

$1 + \frac{\log (2 \sqrt[4]{2^{2}})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Step 7.6

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

$1 + \frac{\log (2 \sqrt{\sqrt{2^{2}}})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Step 7.7

Pull terms out from under the radical, assuming positive real numbers.

$1 + \frac{\log (2 \sqrt{2})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$1 + \frac{\log (2 \sqrt{2})}{\log (2)} + \frac{\log^{2} (\sqrt[4]{8})}{\log^{2} (2)}$

Decimal Form:

$3.0625$