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Chapter 0: Problem 74

Perform the indicated operations. Simplify the result, if possible. $$\frac{1}{x^{2}-2 x-8} \div\left(\frac{1}{x-4}-\frac{1}{x+2}\right)$$

Short Answer

Expert verified
The simplified form of the given expression is \(\frac{1}{6}\).

Step by step solution

01

Express division as multiplication

The expression can be written as multiplication by finding the reciprocal of the divisor, \((\frac{1}{x-4}-\frac{1}{x+2})^{-1}\). So, our expression becomes \(\frac{1}{x^{2}-2 x-8} * (\frac{1}{x-4}-\frac{1}{x+2})^{-1}\).
02

Factorise the denominators in the second fraction

The second fraction can be written on a common denominator. Thus, \(\frac{1}{x-4}-\frac{1}{x+2}= \frac{x+2-x+4}{(x-4)(x+2)}=\frac{6}{x^{2}-2 x-8}\). Thus, our expression can be further written as \(\frac{1}{x^{2}-2 x-8} * (\frac{6}{x^{2}-2 x-8})^{-1}\).
03

Multiply the fractions

Only thing left to do is to multiply the fractions, which simplifies to \(\frac{1}{x^{2}-2 x-8} * \frac{x^{2}-2 x-8}{6} = \frac{1}{6}\).

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