Question

divide as indicated. simplify the answer.\\(\frac{a - b}{3a + 3b} div \frac{a^2 - b^2}{a^2 + 6a + 9}\\)\\(\frac{a - b}{3a + 3b} div \frac{a^2 - b^2}{a^2 + 6a + 9} = square\\)

Explanation:

Step1: Convert division to multiplication

Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. So, we rewrite the division as multiplication:
$$\frac{a - b}{3a + 3b} \div \frac{a^2 - b^2}{a^2 + 6a + 9} = \frac{a - b}{3a + 3b} \times \frac{a^2 + 6a + 9}{a^2 - b^2}$$

Step2: Factor each expression

• Factor \(3a + 3b\): We can factor out a 3, so \(3a + 3b = 3(a + b)\).
• 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 factored forms into the expression, we get:
$$\frac{a - b}{3(a + b)} \times \frac{(a + 3)^2}{(a + b)(a - b)}$$

Step3: Cancel common factors

We can cancel out the common factors of \((a - b)\) in the numerator and denominator, and we also have a common factor that we can analyze. After canceling \((a - b)\), we have:
$$\frac{1}{3(a + b)} \times \frac{(a + 3)^2}{(a + b)}$$

Step4: Multiply the fractions

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

Answer:

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