Question

divide as indicated. simplify the answer.\\(\frac{a - b}{6a + 6b} div \frac{a^2 - b^2}{a^2 + 6a + 9}\\)\\(\frac{a - b}{6a + 6b} div \frac{a^2 - b^2}{a^2 + 6a + 9} = square\\)

Explanation:

Step1: Rewrite division as multiplication

Recall that dividing by a fraction is the same as multiplying by its reciprocal. So, we rewrite the division as multiplication:
$$\frac{a - b}{6a + 6b} \div \frac{a^2 - b^2}{a^2 + 6a + 9} = \frac{a - b}{6a + 6b} \times \frac{a^2 + 6a + 9}{a^2 - b^2}$$

Step2: Factor each expression

• Factor \(6a + 6b\): We can factor out a 6, so \(6a + 6b = 6(a + b)\).
• Factor \(a^2 + 6a + 9\): This is a perfect square trinomial, so \(a^2 + 6a + 9=(a + 3)^2\).
• Factor \(a^2 - b^2\): This is a difference of squares, so \(a^2 - b^2=(a + b)(a - b)\).

Substituting these factorizations into the expression, we get:
$$\frac{a - b}{6(a + b)} \times \frac{(a + 3)^2}{(a + b)(a - b)}$$

Step3: Cancel common factors

We can cancel the common factors of \((a - b)\) in the numerator and denominator, and we are left with:
$$\frac{1}{6(a + b)} \times \frac{(a + 3)^2}{(a + b)}$$

Step4: Multiply the fractions

Multiply the numerators together and the denominators together:
$$\frac{(a + 3)^2}{6(a + b)^2}$$

Answer:

\(\frac{(a + 3)^2}{6(a + b)^2}\)