Questions: Find the quotient and simplify. (r^2-s^2)/(r+s) ÷ r/(r^2+sr) = □ (Simplify your answer.)

Transcript text: Find the quotient and simplify. \[ \frac{r^{2}-s^{2}}{r+s} \div \frac{r}{r^{2}+s r} \] $\frac{r^{2}-s^{2}}{r+s} \div \frac{r}{r^{2}+s r}=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

To solve the given problem, we need to follow these steps:

Recognize that division by a fraction is equivalent to multiplication by its reciprocal.
Simplify the given expression by multiplying the first fraction by the reciprocal of the second fraction.
Factorize the numerator and denominator where possible to simplify the expression further.

Step 1: Rewrite the Expression

We start with the expression: \[ \frac{r^{2}-s^{2}}{r+s} \div \frac{r}{r^{2}+sr} \] This can be rewritten as: \[ \frac{r^{2}-s^{2}}{r+s} \times \frac{r^{2}+sr}{r} \]

Step 2: Simplify the Expression

Next, we multiply the fractions: \[ \frac{(r^{2}-s^{2})(r^{2}+sr)}{r(r+s)} \]

Step 3: Factor and Simplify

The numerator $ r^{2}-s^{2} $ can be factored as $ (r-s)(r+s) $. Thus, we have: \[ \frac{(r-s)(r+s)(r^{2}+sr)}{r(r+s)} \] We can cancel $ (r+s) $ from the numerator and denominator: \[ \frac{(r-s)(r^{2}+sr)}{r} \]