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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 7: Q. 162 (page 680)

In the following exercises, simplify each complex rational expression by writing it as division.
$\frac{\frac{2}{v} + \frac{2}{w}}{\frac{1}{v^{2}} - \frac{1}{w^{2}}}$

Short Answer

Expert verified

The simplified expression is $\frac{2 v w}{w - v}$ .

Step by step solution

01

Step 1. Given Information.

We have:

$\frac{\frac{2}{v} + \frac{2}{w}}{\frac{1}{v^{2}} - \frac{1}{w^{2}}}$

02

Step 2. Simplify the numerator:

The numerator is $\frac{2}{v} + \frac{2}{w}$ .

The least common multiplier of $v, w$ is $v w .$

Adjust the fractions based on the LCM:

$= \frac{2 w}{v w} + \frac{2 v}{v w} = \frac{2 w + 2 v}{v w}$

03

Step 3. Simplify the denominator.

The denominator is $\frac{1}{v^{2}} - \frac{1}{w^{2}}$ .

The least common multiplier of $v^{2}, w^{2}$ is $v^{2} w^{2}$ .

Adjust the fractions based on the LCM:

$= \frac{w^{2}}{v^{2} w^{2}} - \frac{v^{2}}{v^{2} w^{2}} = \frac{w^{2} - v^{2}}{v^{2} w^{2}}$

04

Step 4. Substitute the simplified numerator and denominator into the original expression.

We have:

$\frac{\frac{2}{v} + \frac{2}{w}}{\frac{1}{v^{2}} - \frac{1}{w^{2}}} = \frac{\frac{2 w + 2 v}{v w}}{\frac{w^{2} - v^{2}}{v^{2} w^{2}}}$

Apply fraction rule : $\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a 路鈥� d}{b 路鈥� c}$

$= \frac{(2 w + 2 v) v^{2} w^{2}}{v w (w^{2} - v^{2})} = \frac{2 v w}{w - v}$

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