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Intermediate Algebra

OPENSTAX

$Math Studyset 魅影直播 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 7: Q. 101 (page 667)

In the following exercises, (a) find the LCD for the given rational expressions (b) rewrite them as equivalent rational expressions with the lowest common denominator.
$\frac{9}{z^{2} + 2 z - 8}, \frac{4 z}{z^{2} - 4}$

Short Answer

Expert verified

(a) The LCD is $(z - 2) (z + 2) (z + 4)$

(b) The expressions with equal denominators are $\frac{9 (z + 2)}{(z - 2) (z + 2) (z + 4)}, \frac{4 z (z + 4)}{(z - 2) (z + 2) (z + 4)}$

Step by step solution

01

Part (a) Step 1. Given Information

The given expressions are $\frac{9}{z^{2} + 2 z - 8}, \frac{4 z}{z^{2} - 4}$

02

Part (a) Step 2. Find LCD

Factor the denominators of both the given rational expressions.

$\begin{array}{rcl} z^{2} + 2 z - 8 & = & (z + 4) (z - 2) \\ z^{2} - 4 & = & (z + 2) (z - 2) \end{array}$

So, the LCD is $(z - 2) (z + 2) (z + 4)$ .

03

Part (b) Step 1. Given Information

The given expressions are $\frac{9}{z^{2} + 2 z - 8}, \frac{4 z}{z^{2} - 4}$ .

04

Part (b) Step 2. Express the expressions with LCD

Multiply and divide the expressions by the missing factor to make the denominator equal to LCD.

$\begin{array}{rcl} \frac{9}{z^{2} + 2 z - 8} & = & \frac{9 (z + 2)}{(z - 2) (z + 2) (z + 4)} \\ \frac{4 z}{z^{2} - 4} & = & \frac{4 z (z + 4)}{(z - 2) (z + 2) (z + 4)} \end{array}$

So, the expressions with equal denominators are $\frac{9 (z + 2)}{(z - 2) (z + 2) (z + 4)}, \frac{4 z (z + 4)}{(z - 2) (z + 2) (z + 4)}$

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