Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Apply the distributive property.

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

Step 4

Multiply $3$ by $- 1$ .

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

Step 5

Multiply $- 1$ by $- 2$ .

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

Step 6

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

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

Step 7

Simplify the denominator.

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Step 7.1

Apply the product rule to $(x + 1) \sqrt{x - 1}$ .

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

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

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

Step 7.2.4

Cancel the common factor of $2$ .

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Step 7.2.4.1

Cancel the common factor.

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

Step 7.2.4.2

Rewrite the expression.

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

Step 7.2.5

Simplify.

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

Step 8

Simplify the numerator.

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Step 8.1

Simplify each term.

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Step 8.1.1

Apply the distributive property.

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

Step 8.1.2

Multiply $4$ by $- 3$ .

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

Step 8.1.3

Multiply $2$ by $4$ .

$\frac{\frac{\ln (x - 12 x + 8)}{3}}{{(x + 1)}^{2} (x - 1)}$

Step 8.2

Subtract $12 x$ from $x$ .

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

Step 9

Multiply the numerator by the reciprocal of the denominator.

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

Step 10

Combine.

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

Step 11

Multiply $\ln (- 11 x + 8)$ by $1$ .

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