Chapter 6.2, Problem 37WE

To determine

To calculate: The simplified form of the expression $\frac{a^2-x^2}{a^2}\cdot \frac{a}{3x-3a}$.

Answer to Problem 37WE

The simplified form of the expression $\frac{a^2-x^2}{a^2}\cdot \frac{a}{3x-3a}$ is $-\frac{\left(a+x\right)}{3a}$.

Explanation of Solution

Given information:

The expression $\frac{a^2-x^2}{a^2}\cdot \frac{a}{3x-3a}$.

Formula used:

Fractions are multiplied as $\frac{a}{b}\cdot \frac{c}{d}=\frac{ab}{cd}$.

The difference of square of two numbers a and bis $a^2-b^2=\left(a-b\right)\left(a+b\right)$.

Calculation:

Consider the expression $\frac{a^2-x^2}{a^2}\cdot \frac{a}{3x-3a}$.

Recall that the difference of square of two numbers a and bis $a^2-b^2=\left(a-b\right)\left(a+b\right)$ .

Apply it in the numerator of first term,

  $\begin{array}{c}\frac{a^2-x^2}{a^2}\cdot \frac{a}{3x-3a}=\frac{\left(a-x\right)\left(a+x\right)}{a^2}\cdot \frac{a}{3\left(x-a\right)}\\ =-\frac{\left(a-x\right)\left(a+x\right)}{a^2}\cdot \frac{a}{3\left(a-x\right)}\end{array}$

Strike off the similar terms,

  $\begin{array}{c}\frac{a^2-x^2}{a^2}\cdot \frac{a}{3x-3a}=\frac{\left(a-x\right)\left(a+x\right)}{a^2}\cdot \frac{a}{3\left(x-a\right)}\\ =-\frac{\left(a-x\right)\left(a+x\right)}{a^2}\cdot \frac{a}{3\left(a-x\right)}\\ =-\frac{\left(a+x\right)}{a}\cdot \frac{1}{3}\end{array}$

Recall that fractions are multiplied as $\frac{a}{b}\cdot \frac{c}{d}=\frac{ab}{cd}$.

Apply it,

  $\begin{array}{c}\frac{a^2-x^2}{a^2}\cdot \frac{a}{3x-3a}=\frac{\left(a-x\right)\left(a+x\right)}{a^2}\cdot \frac{a}{3\left(x-a\right)}\\ =-\frac{\left(a-x\right)\left(a+x\right)}{a^2}\cdot \frac{a}{3\left(a-x\right)}\\ =-\frac{\left(a+x\right)}{a}\cdot \frac{1}{3}\\ =-\frac{\left(a+x\right)}{3a}\end{array}$

Thus, the simplified form of the expression $\frac{a^2-x^2}{a^2}\cdot \frac{a}{3x-3a}$ is $-\frac{\left(a+x\right)}{3a}$.