Precalculus Examples

$\frac{\log (10 \sqrt{z} {(x + 5)}^{3})}{7 \sqrt[5]{x - 2}}$

Step 1

Rewrite $\log (10 \sqrt{z} {(x + 5)}^{3})$ as $\log (10 \sqrt{z}) + \log ({(x + 5)}^{3})$ .

$\frac{\log (10 \sqrt{z}) + \log ({(x + 5)}^{3})}{7 \sqrt[5]{x - 2}}$

Step 2

Rewrite $\log (10 \sqrt{z})$ as $\log (10) + \log (\sqrt{z})$ .

$\frac{\log (10) + \log (\sqrt{z}) + \log ({(x + 5)}^{3})}{7 \sqrt[5]{x - 2}}$

Step 3

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

$\frac{\log (10) + \log (z^{\frac{1}{2}}) + \log ({(x + 5)}^{3})}{7 \sqrt[5]{x - 2}}$

Step 4

Expand $\log (z^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{\log (10) + \frac{1}{2} \log (z) + \log ({(x + 5)}^{3})}{7 \sqrt[5]{x - 2}}$

Step 5

Expand $\log ({(x + 5)}^{3})$ by moving $3$ outside the logarithm.

$\frac{\log (10) + \frac{1}{2} \log (z) + 3 \log (x + 5)}{7 \sqrt[5]{x - 2}}$

Step 6

Logarithm base $10$ of $10$ is $1$ .

$\frac{1 + \frac{1}{2} \log (z) + 3 \log (x + 5)}{7 \sqrt[5]{x - 2}}$

Step 7

Simplify the numerator.

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Step 7.1

Combine $\frac{1}{2}$ and $\log (z)$ .

$\frac{1 + \frac{\log (z)}{2} + 3 \log (x + 5)}{7 \sqrt[5]{x - 2}}$

Step 7.2

To write $3 \log (x + 5)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1 + \frac{\log (z)}{2} + 3 \log (x + 5) \cdot \frac{2}{2}}{7 \sqrt[5]{x - 2}}$

Step 7.3

Combine $3 \log (x + 5)$ and $\frac{2}{2}$ .

$\frac{1 + \frac{\log (z)}{2} + \frac{3 \log (x + 5) \cdot 2}{2}}{7 \sqrt[5]{x - 2}}$

Step 7.4

Combine the numerators over the common denominator.

$\frac{1 + \frac{\log (z) + 3 \log (x + 5) \cdot 2}{2}}{7 \sqrt[5]{x - 2}}$

Step 7.5

Multiply $2$ by $3$ .

$\frac{1 + \frac{\log (z) + 6 \log (x + 5)}{2}}{7 \sqrt[5]{x - 2}}$

Step 7.6

Write $1$ as a fraction with a common denominator.

$\frac{\frac{2}{2} + \frac{\log (z) + 6 \log (x + 5)}{2}}{7 \sqrt[5]{x - 2}}$

Step 7.7

Combine the numerators over the common denominator.

$\frac{\frac{2 + \log (z) + 6 \log (x + 5)}{2}}{7 \sqrt[5]{x - 2}}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{1}{7 \sqrt[5]{x - 2}}$

Step 9

Multiply $\frac{1}{7 \sqrt[5]{x - 2}}$ by $\frac{{\sqrt[5]{x - 2}}^{4}}{{\sqrt[5]{x - 2}}^{4}}$ .

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} (\frac{1}{7 \sqrt[5]{x - 2}} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{{\sqrt[5]{x - 2}}^{4}})$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{1}{7 \sqrt[5]{x - 2}}$ by $\frac{{\sqrt[5]{x - 2}}^{4}}{{\sqrt[5]{x - 2}}^{4}}$ .

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{7 \sqrt[5]{x - 2} {\sqrt[5]{x - 2}}^{4}}$

Step 10.2

Move $\sqrt[5]{x - 2}$ .

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{7 (\sqrt[5]{x - 2} {\sqrt[5]{x - 2}}^{4})}$

Step 10.3

Raise $\sqrt[5]{x - 2}$ to the power of $1$ .

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{7 ({\sqrt[5]{x - 2}}^{1} {\sqrt[5]{x - 2}}^{4})}$

Step 10.4

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

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{7 {\sqrt[5]{x - 2}}^{1 + 4}}$

Step 10.5

Add $1$ and $4$ .

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{7 {\sqrt[5]{x - 2}}^{5}}$

Step 10.6

Rewrite ${\sqrt[5]{x - 2}}^{5}$ as $x - 2$ .

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Step 10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x - 2}$ as ${(x - 2)}^{\frac{1}{5}}$ .

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{7 {({(x - 2)}^{\frac{1}{5}})}^{5}}$

Step 10.6.2

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

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{7 {(x - 2)}^{\frac{1}{5} \cdot 5}}$

Step 10.6.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{7 {(x - 2)}^{\frac{5}{5}}}$

Step 10.6.4

Cancel the common factor of $5$ .

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Step 10.6.4.1

Cancel the common factor.

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{7 {(x - 2)}^{\frac{5}{5}}}$

Step 10.6.4.2

Rewrite the expression.

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{7 {(x - 2)}^{1}}$

Step 10.6.5

Simplify.

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{{\sqrt[5]{x - 2}}^{4}}{7 (x - 2)}$

Step 11

Rewrite ${\sqrt[5]{x - 2}}^{4}$ as $\sqrt[5]{{(x - 2)}^{4}}$ .

$\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{\sqrt[5]{{(x - 2)}^{4}}}{7 (x - 2)}$

Step 12

Multiply $\frac{2 + \log (z) + 6 \log (x + 5)}{2} \cdot \frac{\sqrt[5]{{(x - 2)}^{4}}}{7 (x - 2)}$ .

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Step 12.1

Multiply $\frac{2 + \log (z) + 6 \log (x + 5)}{2}$ by $\frac{\sqrt[5]{{(x - 2)}^{4}}}{7 (x - 2)}$ .

$\frac{(2 + \log (z) + 6 \log (x + 5)) \sqrt[5]{{(x - 2)}^{4}}}{2 (7 (x - 2))}$

Step 12.2

Multiply $7$ by $2$ .

$\frac{(2 + \log (z) + 6 \log (x + 5)) \sqrt[5]{{(x - 2)}^{4}}}{14 (x - 2)}$

Step 13

Reorder factors in $\frac{(2 + \log (z) + 6 \log (x + 5)) \sqrt[5]{{(x - 2)}^{4}}}{14 (x - 2)}$ .

$\frac{\sqrt[5]{{(x - 2)}^{4}} (2 + \log (z) + 6 \log (x + 5))}{14 (x - 2)}$