Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

$4 \log (x) \frac{\sqrt[3]{y^{2}} {\sqrt[3]{z}}^{2}}{{\sqrt[3]{z}}^{1 + 2}}$

Step 4.4

Add $1$ and $2$ .

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

$4 \log (x) \frac{\sqrt[3]{y^{2}} {\sqrt[3]{z}}^{2}}{z^{\frac{1}{3} \cdot 3}}$

Step 4.5.3

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

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

Step 4.5.4

Cancel the common factor of $3$ .

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Step 4.5.4.1

Cancel the common factor.

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

Step 4.5.4.2

Rewrite the expression.

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

Step 4.5.5

Simplify.

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

Step 5

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

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

Step 6

Combine using the product rule for radicals.

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

Step 7

Multiply $4 \log (x) \frac{\sqrt[3]{y^{2} z^{2}}}{z}$ .

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Step 7.1

Combine $\frac{\sqrt[3]{y^{2} z^{2}}}{z}$ and $4$ .

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

Step 7.2

Combine $\frac{\sqrt[3]{y^{2} z^{2}} \cdot 4}{z}$ and $\log (x)$ .

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

Step 8

Move $4$ to the left of $\sqrt[3]{y^{2} z^{2}}$ .

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