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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. 8.104 (page 959)

Simplify: $\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a^{2}} - \frac{1}{b^{2}}}$

Short Answer

Expert verified

The simplified form of the given expression is $\frac{a b}{b - a}$ .

Step by step solution

01

Step 1. Given information.

The given expression is:

$\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a^{2}} - \frac{1}{b^{2}}}$

02

Step 2. Concepts.

The following concepts are required for simplification:

$\begin{array}{rcl} \frac{A}{B} 梅 \frac{C}{D} & = & \frac{A}{B} 路 \frac{D}{C} \\ a^{2} - b^{2} & = & (a - b) (a + b) \end{array}$

03

Step 3. Rewrite the expression and then simplify.

Simplify and rewrite the complex fraction as division.

$= \frac{\frac{b + a}{a b}}{\frac{b^{2} - a^{2}}{a^{2} b^{2}}} = \frac{\frac{b + a}{a b}}{\frac{(b - a) (b + a)}{a^{2} b^{2}}} = \frac{b + a}{a b} 梅 \frac{(b - a) (b + a)}{a^{2} b^{2}}$

Rewrite as the product of first times the reciprocal of the second.

$= \frac{b + a}{a b} 路 \frac{a^{2} b^{2}}{(b - a) (b + a)} = \frac{a b}{b - a}$

04

Step 4. Conclusion.

The simplified form of the given expression is $\frac{a b}{b - a}$ .

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