Questions: Divide and, if possible, simplify. (a-5b)/(a^2+10ab+25b^2) ÷ (a^2-25b^2)/(a^2-11ab+30b^2) (a-5b)/(a^2+10ab+25b^2) ÷ (a^2-25b^2)/(a^2-11ab+30b^2) = (Simplify your answer. Type your answer in factored form.)

Solution Steps

We start with the expression:

\[ \frac{a-5 b}{a^{2}+10 a b+25 b^{2}} \div \frac{a^{2}-25 b^{2}}{a^{2}-11 a b+30 b^{2}} \]

This division can be rewritten as multiplication by the reciprocal:

\[ \frac{a-5 b}{a^{2}+10 a b+25 b^{2}} \cdot \frac{a^{2}-11 a b+30 b^{2}}{a^{2}-25 b^{2}} \]

Next, we factor the numerator and denominator of both fractions:

• The numerator of the first fraction \(a - 5b\) remains as is.
• The denominator \(a^{2} + 10ab + 25b^{2}\) factors to \((a + 5b)^{2}\).
• The numerator of the second fraction \(a^{2} - 11ab + 30b^{2}\) factors to \((a - 6b)(a - 5b)\).
• The denominator \(a^{2} - 25b^{2}\) factors to \((a - 5b)(a + 5b)\).

Thus, we have:

\[ \frac{(a - 5b)}{(a + 5b)^{2}} \cdot \frac{(a - 6b)(a - 5b)}{(a - 5b)(a + 5b)} \]

Now we can simplify the expression by canceling out common factors:

\[ \frac{(a - 6b)(a - 5b)}{(a + 5b)^{3}} \]

After canceling \((a - 5b)\) from the numerator and denominator, we arrive at the simplified expression:

\[ \frac{(a - 6b)(a - 5b)}{(a + 5b)^{3}} \]

Final Answer