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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. 172 (page 680)

In the following exercises, simplify each complex rational expression by using the LCD.
$\frac{\frac{1}{m} + \frac{m}{n}}{\frac{n}{m} - \frac{1}{n}}$

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given Information.

We have:

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

02

Step 2. Simplify the numerator.

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

The least common divisor (LCD) is $m n .$

Adjust the fractions based on the LCD:

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

03

Step 3. Simplify the denominator.

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

The least common divisor (LCD) is $m n .$

Adjust the fractions based on the LCD:

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

04

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

We have:

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

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

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

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Most popular questions from this chapter

Solve each rational function inequality and write the solution in interval notation. Given the function $R (x) = \frac{(x - 6)}{(x + 2)}$ , find the values of x that make the function less than or equal to 0.

In the following exercise, perform the indicated operation:

$\frac{6 d}{d - 5} + \frac{1}{d + 4} - \frac{7 d - 5}{d^{2} - d - 20}$

Determine the values for which the rational expression is undefined.

$\frac{5 x^{2} y^{2}}{8 y}$

For the following exercises, find $R (x) = f (x) . g (x)$ where $f (x) a n d g (x)$ are given.

localid="1644679377284" $f (x) = \frac{6 x^{2} - 12 x}{x^{2} + 7 x - 18} a n d g (x) = \frac{x^{2} - 81}{3 x^{2} - 27 x}$

For the following exercises, find $R (x) = f (x) . g (x)$ where $f (x) a n d g (x)$ are given.

$f (x) = \frac{4 x}{x^{2} - 3 x - 10} a n d g (x) = \frac{x^{2} - 25}{8 x^{2}}$

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