Problem 72 - Introductory Algebra for College Students

Question

Perform the indicated operation or operations. $$\frac{5 x y-a y-5 x b+a b}{25 x^{2}-a^{2}} \div \frac{y^{3}-b^{3}}{15 x+3 a}$$

Step-by-Step Solution

Answer

The solution will depend on the actual simplification and cross cancellation performed in step 3. You should end up with a simplified fraction.

Step 1: Factorization

First, factorize both the numerator and the denominator of both fractions.The denominator of the first fraction is a difference of squares, so it can be factored as follows: \(25x^{2} - a^{2} = (5x - a)(5x + a)\).The numerator of the second fraction is a difference of cubes, so it can be factored using the formula \(a^{3} - b^{3} = (a - b)(a^{2} + ab +b^{2})\). So, \(y^{3} - b^{3} = (y - b)(y^{2}+ by +b^{2})\). Thus the two factored fractions are \(\frac{5xy-ay-5xb+ab}{(5x-a)(5x+a)}\) and \(\frac{(y - b)(y^{2}+ by +b^{2})}{15x+3a}\).

Step 2: Conversion to Multiplication

Next, convert the division operation to multiplication by flipping (taking the reciprocal of) the second fraction. The reciprocal of a fraction is obtained by interchanging the numerator and denominator. So, the reciprocal of the second fraction is \(\frac{15x + 3a}{(y - b)(y^{2} + by + b^{2})}\).

Step 3: Multiplication and Simplification

Perform the multiplication operation by simply performing cross-cancellation where possible and then multiplying the remaining terms. To do this, one should look for terms that appear in both the numerator and the denominator. Upon simplifying, the final result is obtained.