Precalculus Examples

$\frac{\ln (\sqrt[3]{x - 1 (3 x - 2) \cdot 4})}{(x + 1) \sqrt{x - 1}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 1 (3 x - 2) \cdot 4}$ as ${(x - 1 (3 x - 2) \cdot 4)}^{\frac{1}{3}}$ .

$\frac{\ln ({(x - 1 (3 x - 2) \cdot 4)}^{\frac{1}{3}})}{(x + 1) \sqrt{x - 1}}$

Step 2

Expand $\ln ({(x - 1 (3 x - 2) \cdot 4)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{\frac{1}{3} \ln (x - 1 (3 x - 2) \cdot 4)}{(x + 1) \sqrt{x - 1}}$

Step 3

Apply the distributive property.

$\frac{\frac{1}{3} \ln (x + (- 1 (3 x) - 1 \cdot - 2) \cdot 4)}{(x + 1) \sqrt{x - 1}}$

Step 4

Multiply $3$ by $- 1$ .

$\frac{\frac{1}{3} \ln (x + (- 3 x - 1 \cdot - 2) \cdot 4)}{(x + 1) \sqrt{x - 1}}$

Step 5

Multiply $- 1$ by $- 2$ .

$\frac{\frac{1}{3} \ln (x + (- 3 x + 2) \cdot 4)}{(x + 1) \sqrt{x - 1}}$

Step 6

Combine $\frac{1}{3}$ and $\ln (x + (- 3 x + 2) \cdot 4)$ .

$\frac{\frac{\ln (x + (- 3 x + 2) \cdot 4)}{3}}{(x + 1) \sqrt{x - 1}}$

Step 7

Simplify the numerator.

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Step 7.1

Simplify each term.

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Step 7.1.1

Apply the distributive property.

$\frac{\frac{\ln (x - 3 x \cdot 4 + 2 \cdot 4)}{3}}{(x + 1) \sqrt{x - 1}}$

Step 7.1.2

Multiply $4$ by $- 3$ .

$\frac{\frac{\ln (x - 12 x + 2 \cdot 4)}{3}}{(x + 1) \sqrt{x - 1}}$

Step 7.1.3

Multiply $2$ by $4$ .

$\frac{\frac{\ln (x - 12 x + 8)}{3}}{(x + 1) \sqrt{x - 1}}$

Step 7.2

Subtract $12 x$ from $x$ .

$\frac{\frac{\ln (- 11 x + 8)}{3}}{(x + 1) \sqrt{x - 1}}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{1}{(x + 1) \sqrt{x - 1}}$

Step 9

Multiply $\frac{1}{(x + 1) \sqrt{x - 1}}$ by $\frac{\sqrt{x - 1}}{\sqrt{x - 1}}$ .

$\frac{\ln (- 11 x + 8)}{3} (\frac{1}{(x + 1) \sqrt{x - 1}} \cdot \frac{\sqrt{x - 1}}{\sqrt{x - 1}})$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{1}{(x + 1) \sqrt{x - 1}}$ by $\frac{\sqrt{x - 1}}{\sqrt{x - 1}}$ .

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) \sqrt{x - 1} \sqrt{x - 1}}$

Step 10.2

Move $\sqrt{x - 1}$ .

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) (\sqrt{x - 1} \sqrt{x - 1})}$

Step 10.3

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

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) ({\sqrt{x - 1}}^{1} \sqrt{x - 1})}$

Step 10.4

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

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) ({\sqrt{x - 1}}^{1} {\sqrt{x - 1}}^{1})}$

Step 10.5

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

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) {\sqrt{x - 1}}^{1 + 1}}$

Step 10.6

Add $1$ and $1$ .

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) {\sqrt{x - 1}}^{2}}$

Step 10.7

Rewrite ${\sqrt{x - 1}}^{2}$ as $x - 1$ .

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Step 10.7.1

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

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) {({(x - 1)}^{\frac{1}{2}})}^{2}}$

Step 10.7.2

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

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) {(x - 1)}^{\frac{1}{2} \cdot 2}}$

Step 10.7.3

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

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) {(x - 1)}^{\frac{2}{2}}}$

Step 10.7.4

Cancel the common factor of $2$ .

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Step 10.7.4.1

Cancel the common factor.

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) {(x - 1)}^{\frac{2}{2}}}$

Step 10.7.4.2

Rewrite the expression.

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) {(x - 1)}^{1}}$

Step 10.7.5

Simplify.

$\frac{\ln (- 11 x + 8)}{3} \cdot \frac{\sqrt{x - 1}}{(x + 1) (x - 1)}$

Step 11

Multiply $\frac{\ln (- 11 x + 8)}{3}$ by $\frac{\sqrt{x - 1}}{(x + 1) (x - 1)}$ .

$\frac{\ln (- 11 x + 8) \sqrt{x - 1}}{3 (x + 1) (x - 1)}$