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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. 278 (page 966)

In the following exercises, simplify:
$\frac{\frac{2}{r} + \frac{2}{t}}{\frac{1}{r^{2}} - \frac{1}{t^{2}}}$ .

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given Information.

The rational expression is,

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

02

Step 2. Simplifying the rational expression.

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

The LCD of the expression is $r^{2} t^{2}$ .

$= \frac{r^{2} t^{2} (\frac{2}{r} + \frac{2}{t})}{r^{2} t^{2} (\frac{1}{r^{2}} - \frac{1}{t^{2}})} = \frac{r^{2} t^{2} 脳 \frac{2}{r} + r^{2} t^{2} 脳 \frac{2}{t}}{r^{2} t^{2} 脳 \frac{1}{r^{2}} - r^{2} t^{2} 脳 \frac{1}{t^{2}}} = \frac{2 r t^{2} + 2 r^{2} t}{t^{2} - r^{2}}$

Using $(a^{2} - b^{2}) = (a + b) (a - b)$ in the denominator,

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

Removing common factors,

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

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