Precalculus Examples

${(\frac{1}{x^{2} - 5 x - 14})}^{\frac{1}{2}}$

Step 1

Factor $x^{2} - 5 x - 14$ 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 $- 14$ and whose sum is $- 5$ .

$- 7, 2$

Step 1.2

Write the factored form using these integers.

${(\frac{1}{(x - 7) (x + 2)})}^{\frac{1}{2}}$

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 $\frac{1}{(x - 7) (x + 2)}$ .

$\frac{1^{\frac{1}{2}}}{{((x - 7) (x + 2))}^{\frac{1}{2}}}$

Step 2.2

Apply the product rule to $(x - 7) (x + 2)$ .

$\frac{1^{\frac{1}{2}}}{{(x - 7)}^{\frac{1}{2}} {(x + 2)}^{\frac{1}{2}}}$

Step 3

One to any power is one.

$\frac{1}{{(x - 7)}^{\frac{1}{2}} {(x + 2)}^{\frac{1}{2}}}$