Precalculus Examples

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

Step 1

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

$2 \ln (x) \sqrt{\frac{y}{z}}$

Step 2

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

$2 \ln (x) \frac{\sqrt{y}}{\sqrt{z}}$

Step 3

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

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

$2 \ln (x) \frac{\sqrt{y} \sqrt{z}}{\sqrt{z} \sqrt{z}}$

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Simplify.

$2 \ln (x) \frac{\sqrt{y} \sqrt{z}}{z}$

Step 5

Combine using the product rule for radicals.

$2 \ln (x) \frac{\sqrt{y z}}{z}$

Step 6

Multiply $2 \ln (x) \frac{\sqrt{y z}}{z}$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Move $2$ to the left of $\sqrt{y z}$ .

$\frac{2 \sqrt{y z} \ln (x)}{z}$