Problem 61 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \( (2x - 2y)^2 \).

Step 2: Substitution

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

Step 3: 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 \( \frac{(x + y)(x - y)}{(2x - 2y)^2} \times 4 \frac{x + y}{x - y} \)

Step 4: Simplify

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

Step 5: Further Simplify

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