Precalculus Examples

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

Step 1

Expand $\log_{2} (x^{4})$ by moving $4$ outside the logarithm.

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

Step 2

Rewrite $\frac{y}{z^{3}}$ as ${(\frac{1}{z})}^{2} \frac{y}{z}$ .

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Step 2.1

Factor the perfect power $1^{2}$ out of $y$ .

$4 \log_{2} (x) \sqrt{\frac{1^{2} y}{z^{3}}}$

Step 2.2

Factor the perfect power $z^{2}$ out of $z^{3}$ .

$4 \log_{2} (x) \sqrt{\frac{1^{2} y}{z^{2} z}}$

Step 2.3

Rearrange the fraction $\frac{1^{2} y}{z^{2} z}$ .

$4 \log_{2} (x) \sqrt{{(\frac{1}{z})}^{2} \frac{y}{z}}$

Step 3

Pull terms out from under the radical.

$4 \log_{2} (x) (\frac{1}{z} \sqrt{\frac{y}{z}})$

Step 4

Rewrite $\sqrt{\frac{y}{z}}$ as $\frac{\sqrt{y}}{\sqrt{z}}$ .

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y}}{\sqrt{z}})$

Step 5

Multiply $\frac{\sqrt{y}}{\sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

$4 \log_{2} (x) (\frac{1}{z} (\frac{\sqrt{y}}{\sqrt{z}} \cdot \frac{\sqrt{z}}{\sqrt{z}}))$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt{y}}{\sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y} \sqrt{z}}{\sqrt{z} \sqrt{z}})$

Step 6.2

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

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y} \sqrt{z}}{{\sqrt{z}}^{1} \sqrt{z}})$

Step 6.3

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

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y} \sqrt{z}}{{\sqrt{z}}^{1} {\sqrt{z}}^{1}})$

Step 6.4

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

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y} \sqrt{z}}{{\sqrt{z}}^{1 + 1}})$

Step 6.5

Add $1$ and $1$ .

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y} \sqrt{z}}{{\sqrt{z}}^{2}})$

Step 6.6

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

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Step 6.6.1

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

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y} \sqrt{z}}{{(z^{\frac{1}{2}})}^{2}})$

Step 6.6.2

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

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y} \sqrt{z}}{z^{\frac{1}{2} \cdot 2}})$

Step 6.6.3

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

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y} \sqrt{z}}{z^{\frac{2}{2}}})$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y} \sqrt{z}}{z^{\frac{2}{2}}})$

Step 6.6.4.2

Rewrite the expression.

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y} \sqrt{z}}{z^{1}})$

Step 6.6.5

Simplify.

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y} \sqrt{z}}{z})$

Step 7

Combine using the product rule for radicals.

$4 \log_{2} (x) (\frac{1}{z} \cdot \frac{\sqrt{y z}}{z})$

Step 8

Multiply $\frac{1}{z} \cdot \frac{\sqrt{y z}}{z}$ .

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Step 8.1

Multiply $\frac{1}{z}$ by $\frac{\sqrt{y z}}{z}$ .

$4 \log_{2} (x) \frac{\sqrt{y z}}{z \cdot z}$

Step 8.2

Raise $z$ to the power of $1$ .

$4 \log_{2} (x) \frac{\sqrt{y z}}{z^{1} z}$

Step 8.3

Raise $z$ to the power of $1$ .

$4 \log_{2} (x) \frac{\sqrt{y z}}{z^{1} z^{1}}$

Step 8.4

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

$4 \log_{2} (x) \frac{\sqrt{y z}}{z^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$4 \log_{2} (x) \frac{\sqrt{y z}}{z^{2}}$

Step 9

Multiply $4 \log_{2} (x) \frac{\sqrt{y z}}{z^{2}}$ .

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Step 9.1

Combine $\frac{\sqrt{y z}}{z^{2}}$ and $4$ .

$\frac{\sqrt{y z} \cdot 4}{z^{2}} \log_{2} (x)$

Step 9.2

Combine $\frac{\sqrt{y z} \cdot 4}{z^{2}}$ and $\log_{2} (x)$ .

$\frac{\sqrt{y z} \cdot 4 \log_{2} (x)}{z^{2}}$

Step 10

Move $4$ to the left of $\sqrt{y z}$ .

$\frac{4 \sqrt{y z} \log_{2} (x)}{z^{2}}$