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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. 159 (page 680)

In the following exercises, simplify each complex rational expression by writing it as division.
$\frac{\frac{n 鈥�}{m} + \frac{1}{n}}{\frac{1}{n} - \frac{n}{m} 鈥�}$

Short Answer

Expert verified

The simplified expression is $\frac{n^{2} + m}{m - n^{2}}$ .

Step by step solution

01

Step 1. Given Information.

We have:

$\frac{\frac{n 鈥�}{m} + \frac{1}{n}}{\frac{1}{n} - \frac{n}{m} 鈥�}$

02

Step 2. Simplify the numerator:

The numerator is $\frac{n 鈥�}{m} + \frac{1}{n}$ .

The least common multiplier of $m, n$ is $m n .$

Adjust the fractions based on the LCM:

$= \frac{n^{2}}{n m} + \frac{m}{n m} = \frac{n^{2} + m}{n m}$

03

Step 3. Simplify the denominator.

The denominator is $\frac{1}{n} - \frac{n}{m}$ .

The least common multiplier of $n, m$ is $n m .$

$= \frac{m}{n m} - \frac{n^{2}}{n m} = \frac{m - n^{2}}{n m}$

04

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

We have:

$\frac{\frac{n 鈥�}{m} + \frac{1}{n}}{\frac{1}{n} - \frac{n}{m} 鈥�} = \frac{\frac{n^{2} + m}{n m}}{\frac{m - n^{2}}{n m}}$

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

$= \frac{(n^{2} + m) n m}{n m (m - n^{2})} = \frac{n^{2} + m}{m - n^{2}}$

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