Precalculus Examples

$\log_{6} (x^{9}) \sqrt{\frac{y}{z^{7}}}$

Step 1

Expand $\log_{6} (x^{9})$ by moving $9$ outside the logarithm.

$9 \log_{6} (x) \sqrt{\frac{y}{z^{7}}}$

Step 2

Rewrite $\frac{y}{z^{7}}$ as ${(\frac{1}{z^{3}})}^{2} \frac{y}{z}$ .

Tap for more steps...

Step 2.1

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

$9 \log_{6} (x) \sqrt{\frac{1^{2} y}{z^{7}}}$

Step 2.2

Factor the perfect power ${(z^{3})}^{2}$ out of $z^{7}$ .

$9 \log_{6} (x) \sqrt{\frac{1^{2} y}{{(z^{3})}^{2} z}}$

Step 2.3

Rearrange the fraction $\frac{1^{2} y}{{(z^{3})}^{2} z}$ .

$9 \log_{6} (x) \sqrt{{(\frac{1}{z^{3}})}^{2} \frac{y}{z}}$

Step 3

Pull terms out from under the radical.

$9 \log_{6} (x) (\frac{1}{z^{3}} \sqrt{\frac{y}{z}})$

Step 4

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

$9 \log_{6} (x) (\frac{1}{z^{3}} \cdot \frac{\sqrt{y}}{\sqrt{z}})$

Step 5

Combine.

$9 \log_{6} (x) \frac{1 \sqrt{y}}{z^{3} \sqrt{z}}$

Step 6

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

$9 \log_{6} (x) \frac{\sqrt{y}}{z^{3} \sqrt{z}}$

Step 7

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

$9 \log_{6} (x) (\frac{\sqrt{y}}{z^{3} \sqrt{z}} \cdot \frac{\sqrt{z}}{\sqrt{z}})$

Step 8

Combine and simplify the denominator.

Tap for more steps...

Step 8.1

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

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} \sqrt{z} \sqrt{z}}$

Step 8.2

Move $\sqrt{z}$ .

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} (\sqrt{z} \sqrt{z})}$

Step 8.3

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

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} ({\sqrt{z}}^{1} \sqrt{z})}$

Step 8.4

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

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} ({\sqrt{z}}^{1} {\sqrt{z}}^{1})}$

Step 8.5

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

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} {\sqrt{z}}^{1 + 1}}$

Step 8.6

Add $1$ and $1$ .

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} {\sqrt{z}}^{2}}$

Step 8.7

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

Tap for more steps...

Step 8.7.1

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

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} {(z^{\frac{1}{2}})}^{2}}$

Step 8.7.2

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

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} z^{\frac{1}{2} \cdot 2}}$

Step 8.7.3

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

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} z^{\frac{2}{2}}}$

Step 8.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.7.4.1

Cancel the common factor.

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} z^{\frac{2}{2}}}$

Step 8.7.4.2

Rewrite the expression.

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} z^{1}}$

Step 8.7.5

Simplify.

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} z}$

Step 9

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

Tap for more steps...

Step 9.1

Multiply $z^{3}$ by $z$ .

Tap for more steps...

Step 9.1.1

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

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3} z^{1}}$

Step 9.1.2

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

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{3 + 1}}$

Step 9.2

Add $3$ and $1$ .

$9 \log_{6} (x) \frac{\sqrt{y} \sqrt{z}}{z^{4}}$

Step 10

Combine using the product rule for radicals.

$9 \log_{6} (x) \frac{\sqrt{y z}}{z^{4}}$

Step 11

Multiply $9 \log_{6} (x) \frac{\sqrt{y z}}{z^{4}}$ .

Tap for more steps...

Step 11.1

Combine $\frac{\sqrt{y z}}{z^{4}}$ and $9$ .

$\frac{\sqrt{y z} \cdot 9}{z^{4}} \log_{6} (x)$

Step 11.2

Combine $\frac{\sqrt{y z} \cdot 9}{z^{4}}$ and $\log_{6} (x)$ .

$\frac{\sqrt{y z} \cdot 9 \log_{6} (x)}{z^{4}}$

Step 12

Move $9$ to the left of $\sqrt{y z}$ .

$\frac{9 \sqrt{y z} \log_{6} (x)}{z^{4}}$