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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. 174 (page 680)

In the following exercises, simplify each complex rational expression by using the LCD.
$\frac{\frac{2}{r} + \frac{2}{t}}{鈥� \frac{1}{r^{2}} - \frac{1}{t^{2}}}$

Short Answer

Expert verified

The simplified expression is $\frac{2 r t}{t - r}$ .

Step by step solution

01

Step 1. Given Information.

We have:

$\frac{\frac{2}{r} + \frac{2}{t}}{鈥� \frac{1}{r^{2}} - \frac{1}{t^{2}}}$

02

Step 2. Simplify the numerator.

The numerator is $\frac{2}{r} + \frac{2}{t}$ .

The least common divisor (LCD) is $r t .$

Adjust the fractions based on the LCD:

$= \frac{2 t}{r t} + \frac{2 r}{r t} = \frac{2 t + 2 r}{r t}$

03

Step 3. Simplify the denominator.

The denominator is $\frac{1}{r^{2}} - \frac{1}{t^{2}}$ .

The least common divisor (LCD) is $r^{2} t^{2}$ .

Adjust the fractions based on the LCD:

$= \frac{t^{2}}{r^{2} t^{2}} - \frac{r^{2}}{r^{2} t^{2}} = \frac{t^{2} - r^{2}}{r^{2} t^{2}}$

04

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

We have:

$\frac{\frac{2}{r} + \frac{2}{t}}{鈥� \frac{1}{r^{2}} - \frac{1}{t^{2}}} = \frac{\frac{2 t + 2 r}{r t}}{\frac{t^{2} - r^{2}}{r^{2} t^{2}}}$

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

$= \frac{(2 t + 2 r) r^{2} t^{2}}{r t (t^{2} - r^{2})} = \frac{2 r t}{t - r}$

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Most popular questions from this chapter

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