Chapter 8: Q. 269 (page 966)

In the following exercises, simplify.
$\frac{2 - \frac{2}{a + 3}}{\frac{1}{a + 3} + \frac{a}{2}}$

Short Answer

Expert verified

The simplified form of the given expression is $\frac{4}{a + 1} .$

Step by step solution

Step 1. Given information.

The given expression is:

$\frac{2 - \frac{2}{a + 3}}{\frac{1}{a + 3} + \frac{a}{2}}$

Step 2. Concept.

The following concept is required for simplification:

$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}$

Step 3. Rewrite the expression.

Simplify the numerator and denominator.

$= \frac{\frac{2 (a + 3) - 2}{a + 3}}{\frac{2 + a (a + 3)}{2 (a + 3)}} = \frac{\frac{2 a + 6 - 2}{a + 3}}{\frac{2 + a^{2} + 3 a)}{2 (a + 3)}} = \frac{\frac{2 a + 4}{a + 3}}{\frac{a^{2} + 3 a + 2}{2 (a + 3)}} = \frac{2 a + 4}{a + 3} \div \frac{a^{2} + 3 a + 2}{2 (a + 3)}$

Step 4. Simplify the expression.

Rewrite as the product of first times the reciprocal of the second. $= \frac{2 a + 4}{a + 3} \cdot \frac{2 (a + 3)}{a^{2} + 3 a + 2} = \frac{2 a + 4}{a + 3} \cdot \frac{2 (a + 3)}{a^{2} + 2 a + a + 2} = \frac{2 (a + 2)}{a + 3} \cdot \frac{2 (a + 3)}{a (a + 2) + 1 (a + 2)} = \frac{2 (a + 2)}{a + 3} \cdot \frac{2 (a + 3)}{(a + 2) (a + 1)} = \frac{4}{a + 1}$

Step 5. Conclusion.

The simplified form of the given expression is $\frac{4}{a + 1}$ .

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In the following exercises, simplify.
$\frac{2 - \frac{2}{a + 3}}{\frac{1}{a + 3} + \frac{a}{2}}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept.

Step 3. Rewrite the expression.

Step 4. Simplify the expression.

Step 5. Conclusion.

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In the following exercises, simplify.2-2a+31a+3+a2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concept.

Step 3. Rewrite the expression.

Step 4. Simplify the expression.

Step 5. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Theoretical and Mathematical Physics

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

In the following exercises, simplify.
$\frac{2 - \frac{2}{a + 3}}{\frac{1}{a + 3} + \frac{a}{2}}$