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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. 255 (page 966)

In the following exercises, simplify.
$\frac{\frac{2 a}{a + 4}}{\frac{4 a^{2}}{a^{2} - 16}}$

Short Answer

Expert verified

The simplified form of the given expression is $\frac{a - 4}{2 a}$ .

Step by step solution

01

Step 1. Given information.

The given expression is:

$\frac{\frac{2 a}{a + 4}}{\frac{4 a^{2}}{a^{2} - 16}}$

02

Step 2. Concepts.

Some 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.

Rewrite the complex fraction as division.

$\frac{2 a}{a + 4} 梅 \frac{4 a^{2}}{a^{2} - 16}$

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

$\begin{array}{rcl} = & \frac{2 a}{a + 4} 路 \frac{a^{2} - 16}{4 a^{2}} \\ = & \frac{2 a (a^{2} - 16)}{(a + 4) 4 a^{2}} \\ = & \frac{2 a (a - 4) (a + 4)}{(a + 4) (2 a) (2 a)} \\ = & \frac{a - 4}{2 a} \end{array}$

04

Step 4. Conclusion.

The simplified form of the given expression is $\frac{a - 4}{2 a}$ .

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