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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. 160 (page 680)

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

Short Answer

Expert verified

The simplified expression is $\frac{q + p^{2}}{q (q - 1)}$ .

Step by step solution

01

Step 1. Given Information.

We have:

$\frac{\frac{1}{p} + \frac{p}{q}}{\frac{q}{p} - \frac{1}{p}}$

02

Step 2. Simplify the numerator:

The numerator is $\frac{1}{p} + \frac{p}{q}$ .

The least common multiplier of $p, q$ is $p q .$

Adjust the fractions based on the LCM:

$= \frac{q}{p q} + \frac{p^{2}}{p q} = \frac{q + p^{2}}{p q}$

03

Step 3. Simplify the denominator.

The denominator is $\frac{q}{p} - \frac{1}{p}$ .

The least common multiplier of $p, p$ is $p .$

Adjust the fractions based on the LCM:

$= \frac{q - 1}{p}$

04

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

We have:

$\frac{\frac{1}{p} + \frac{p}{q}}{\frac{q}{p} - \frac{1}{p}} = \frac{\frac{q + p^{2}}{p q}}{\frac{q - 1}{p}}$

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

$= \frac{(q + p^{2}) p}{p q (q - 1)} = \frac{q + p^{2}}{q (q - 1)}$

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