QuestionDivide as indicated. Simplify the answer. $\begin{array}{r} \frac{a-b}{5 a+5 b} \div \frac{a^{2}-b^{2}}{a^{2}+4 a+4} \\ \frac{a-b}{5 a+5 b} \div \frac{a^{2}-b^{2}}{a^{2}+4 a+4}=\square \end{array}$ $\square$

Studdy Solution

STEP 1

1. The problem involves division of two rational expressions.
2. Simplification involves factoring polynomials and reducing common factors.
3. The expression is simplified by multiplying by the reciprocal.

STEP 2

1. Rewrite the division as multiplication by the reciprocal.
2. Factor the polynomials in the numerators and denominators.
3. Simplify by canceling common factors.

STEP 3

Rewrite the division of fractions as multiplication by the reciprocal:
$\frac{a-b}{5a+5b} \div \frac{a^2-b^2}{a^2+4a+4} = \frac{a-b}{5a+5b} \times \frac{a^2+4a+4}{a^2-b^2}$

STEP 4

Factor the polynomials:
- The denominator $5a + 5b$ can be factored as $5(a + b)$ . - The numerator $a^2 - b^2$ is a difference of squares and can be factored as $(a-b)(a+b)$ . - The numerator $a^2 + 4a + 4$ is a perfect square trinomial and can be factored as $(a+2)^2$ .
The expression becomes:
$\frac{a-b}{5(a+b)} \times \frac{(a+2)^2}{(a-b)(a+b)}$

STEP 5

Simplify the expression by canceling common factors:
- Cancel $a-b$ from the numerator and denominator. - Cancel $a+b$ from the numerator and denominator.
The simplified expression is:
$\frac{(a+2)^2}{5}$
The simplified answer is:
$\boxed{\frac{(a+2)^2}{5}}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms