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Chapter 7: Problem 61

Divide as indicated. $$\frac{x^{2}-y^{2}}{8 x^{2}-16 x y+8 y^{2}} \div \frac{4 x-4 y}{x+y}$$

Short Answer

Expert verified
The simplified form of the given expression is \( \frac{x + y}{x - y} \)

Step by step solution

01

Analyze Given Fractions

The given problem consists of two fractions. The second fraction \( \frac{4 x-4 y}{x+y} \)can be simplified by factoring 4 from the numerator, yielding \( 4 \frac{x - y}{x + y} \). For the first fraction, the numerator is a difference of squares and can be factored as \( (x + y)(x - y) \) and the denominator can be factored by noticing it's a perfect square trinomial as \( (2x - 2y)^2 \).
02

Substitution

After factoring, the first fraction becomes \( \frac{(x + y)(x - y)}{(2x - 2y)^2} \) and the second becomes \( 4 \frac{x - y}{x + y} \)
03

Swap and Divide

The process of division by a fraction can be converted into multiplication by its reciprocal. This makes \( \frac{(x + y)(x - y)}{(2x - 2y)^2} \div 4 \frac{x - y}{x + y} \) convert to \( \frac{(x + y)(x - y)}{(2x - 2y)^2} \times 4 \frac{x + y}{x - y} \)
04

Simplify

Simplify by cancelling out the common factors from the numerator and the denominator. This leaves \( \frac{2(x + y)}{2x - 2y} \)
05

Further Simplify

Simplify by factoring 2 from the denominator, resulting in \( \frac{x + y}{x - y} \)

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Most popular questions from this chapter

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{3-x}{x-7}-\frac{2 x-5}{7-x}$$

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Use a proportion to solve each problem. A person's hair length is proportional to the number of years it has been growing. After 2 years, a person's hair grows 8 inches. The longest moustache on record was grown by Kalyan Sain of India. Sain grew his moustache for 17 years. How long was each side of the moustache?

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Factor: \(25 x^{2}-81 .\) (Section 6.4, Example 1)
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