Precalculus Examples

$\ln ({(\frac{x^{2} - x - 2}{{(x + 4)}^{5}})}^{\frac{1}{3}})$

Step 1

Factor $x^{2} - x - 2$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.2

Write the factored form using these integers.

$\ln ({(\frac{(x - 2) (x + 1)}{{(x + 4)}^{5}})}^{\frac{1}{3}})$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1

Apply the product rule to $\frac{(x - 2) (x + 1)}{{(x + 4)}^{5}}$ .

$\ln (\frac{{((x - 2) (x + 1))}^{\frac{1}{3}}}{{({(x + 4)}^{5})}^{\frac{1}{3}}})$

Step 2.2

Apply the product rule to $(x - 2) (x + 1)$ .

$\ln (\frac{{(x - 2)}^{\frac{1}{3}} {(x + 1)}^{\frac{1}{3}}}{{({(x + 4)}^{5})}^{\frac{1}{3}}})$

Step 3

Multiply the exponents in ${({(x + 4)}^{5})}^{\frac{1}{3}}$ .

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Step 3.1

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

$\ln (\frac{{(x - 2)}^{\frac{1}{3}} {(x + 1)}^{\frac{1}{3}}}{{(x + 4)}^{5 (\frac{1}{3})}})$

Step 3.2

Combine $5$ and $\frac{1}{3}$ .

$\ln (\frac{{(x - 2)}^{\frac{1}{3}} {(x + 1)}^{\frac{1}{3}}}{{(x + 4)}^{\frac{5}{3}}})$