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Elementary Algebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2nd Edition 路 1240 pages

ISBN: 9780998625713

Chapter 8: Q. 575 (page 1027)

In the following exercises, simplify.
$\frac{\frac{2}{x^{2} - 49} + \frac{1}{z + 7}}{\frac{9}{z + 7} + \frac{12}{z - 7}}$

Short Answer

Expert verified

The simplification of the expression $\frac{\frac{2}{x^{2} - 49} + \frac{1}{z + 7}}{\frac{9}{z + 7} + \frac{12}{z - 7}}$ is $\frac{z - 5}{11 z + 21}$

Step by step solution

01

Step 1. Given Information.

An expression,

$\frac{\frac{2}{x^{2} - 49} + \frac{1}{z + 7}}{\frac{9}{z + 7} + \frac{12}{z - 7}}$

02

Step 2. Find the LCD of all fractions in the complex rational expression.

The LCD of the given expression is, $(z + 7) (z - 7)$

03

Step 3. Multiply the numerator and denominator by the LCD.

$\frac{\frac{2}{x^{2} - 49} + \frac{1}{z + 7}}{\frac{9}{z + 7} + \frac{12}{z - 7}} = (\frac{\frac{2}{x^{2} - 49} + \frac{1}{z + 7}}{\frac{9}{z + 7} + \frac{12}{z - 7}}) \frac{(z + 7) (z - 7)}{(z + 7) (z - 7)}$

04

Step 4. Simplify the expression.

$\frac{\frac{2}{x^{2} - 49} + \frac{1}{z + 7}}{\frac{9}{z + 7} + \frac{12}{z - 7}} = \frac{2 + (z - 7)}{9 (z - 7) + 12 (z + 7)} = \frac{z - 5}{9 z - 63 + 2 z + 84} = \frac{z - 5}{11 z + 21}$

Notice that there are no more factors common to the numerator and denominator.

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