Problem 71 - Introductory Algebra for College Students

Question

Perform the indicated operation or operations. $$\frac{3 x y+a y+3 x b+a b}{9 x^{2}-a^{2}} \div \frac{y^{3}+b^{3}}{6 x-2 a}$$

Step-by-Step Solution

Answer

The desired result will be the simplified form of the expression \( \frac{(3 x y+a y)(6x-2a) + (3x b + a b)(6x -2a)}{ (9x^{2}-a^{2})(y^{3}+b^{3})} \). Without given values for the variables, we cannot simplify further. The final expression may vary based on the given terms.

Step 1: Identify the reciprocal of the second fraction

Recall that the reciprocal of a fraction is obtained by swapping the numerator and the denominator. Therefore, the reciprocal of \( \frac{y^{3}+b^{3}}{6 x-2 a} \) is \( \frac{6 x-2 a}{y^{3}+b^{3}} \).

Step 2: Rewrite the division operation as multiplication

Now, replace the division operation with multiplication, and multiply the first fraction by the reciprocal of the second fraction: \( \frac{3 x y+a y+3 x b+a b}{9 x^{2}-a^{2}} \times \frac{6 x-2 a}{y^{3}+b^{3}} \).

Step 3: Apply the distributive law

Apply the distributive law (a(b + c) = ab + ac) to the numerator:\( \frac{(3 x y+a y)(6x-2a) + (3x b + a b)(6x -2a)}{ (9x^{2}-a^{2})(y^{3}+b^{3})} \)

Step 4: Simplification

Perform multiplication and combine like terms in the numerator and denominator. The expression will be your final answer after simplification.