Chapter 5.5, Problem 22WE

To determine

To simplify the given expression.

Answer to Problem 22WE

  $1$

Explanation of Solution

Given:

Expression: $\frac{r^2+4rs+3s^2}{r^2+5rs+6s^2}\cdot {\left(r+2s\right)}^{-1}÷\frac{r+s}{r^2+4rs+4s^2}$

Concept used:

To multiply two fractions, simply multiply numerators by each other and denominators by each other and then reduce the fraction if possible.

  $\frac{a}{b}\cdot \frac{c}{d}\to \frac{a\times c}{b\times d}$

To divide two fractions, simply flip the second fraction into a reciprocal and change the division sign to a multiplication sign and then multiply the two fractions by multiplying the numerators by each other and denominators by each other. As illustrated below:

  $\frac{a}{b}÷\frac{c}{d}\to \frac{a}{b}\times \frac{d}{c}\to \frac{a\times d}{b\times c}$

Calculation:

Consider the given expression,

  $\Rightarrow \frac{r^2+4rs+3s^2}{r^2+5rs+6s^2}\cdot {\left(r+2s\right)}^{-1}÷\frac{r+s}{r^2+4rs+4s^2}$

  $\Rightarrow \frac{r^2+4rs+3s^2}{r^2+5rs+6s^2}\cdot \frac{1}{\left(r+2s\right)}\cdot \frac{r^2+4rs+4s^2}{r+s}$   [Flipping second and third fractions]

  $\Rightarrow \frac{r^2+rs+3rs+3s^2}{r^2+2rs+3rs+6s^2}\times \frac{1}{\left(r+2s\right)}\times \frac{r^2+2rs+2rs+4s^2}{r+s}$  [Factorizing the fractions]

  $\Rightarrow \frac{r\left(r+s\right)+3s\left(r+s\right)}{r\left(r+2s\right)+3s\left(r+2s\right)}\times \frac{1}{\left(r+2s\right)}\times \frac{r\left(r+2s\right)+2s\left(r+2s\right)}{r+s}$

  $\Rightarrow \frac{\left(r+s\right)\left(r+3s\right)}{\left(r+2s\right)\left(r+3s\right)}\times \frac{1}{\left(r+2s\right)}\times \frac{\left(r+2s\right)\left(r+2s\right)}{r+s}$

  $\Rightarrow \frac{1}{1}\times \frac{1}{1}\times \frac{1}{1}$  [Reducing the fractions]

  $\Rightarrow 1$

Conclusion:

Therefore, after simplification, the answer is $1$ .