Chapter 6.5, Problem 58WE

To determine

Find the simplified form of the given expression.

Answer to Problem 58WE

The simplified form of the expression is $-\frac{3}{cd}$ .

Explanation of Solution

Given:

The given expression is $\frac{4}{c^2-4cd}-\frac{1}{cd-4d^2}-\frac{2}{cd}$ .

Calculation:

Rewrite the given expression.

  $\begin{array}{l}\frac{4}{c^2-4cd}-\frac{1}{cd-4d^2}-\frac{2}{cd}\\ =\frac{4}{c\left(c-4d\right)}-\frac{1}{d\left(c-4d\right)}-\frac{2}{cd}\end{array}$

Since the denominators are different, rewrite the fractions using their least common denominator, $cd\left(c-4d\right)$ .

  $\begin{array}{l}=\frac{4d}{cd\left(c-4d\right)}-\frac{c}{cd\left(c-4d\right)}-\frac{2\left(c-4d\right)}{cd\left(c-4d\right)}\\ =\frac{4d}{cd\left(c-4d\right)}-\frac{c}{cd\left(c-4d\right)}-\frac{2\left(c-4d\right)}{cd\left(c-4d\right)}\\ =\frac{4d}{cd\left(c-4d\right)}-\frac{c}{cd\left(c-4d\right)}-\frac{\left(2c-8d\right)}{cd\left(c-4d\right)}\end{array}$

Use addition/subtraction rule of fraction.

  $\frac{a}{c}\pm \frac{b}{c}=\frac{a\pm b}{c}$

To add/subtract fractions with the same denominator, add/subtract their numerators and write the result over the common denominator.

  $\begin{array}{l}=\frac{4d-c-2c+8d}{cd\left(c-4d\right)}\\ =\frac{-3c+12d}{cd\left(c-4d\right)}\\ =\frac{-3\left(c-4d\right)}{cd\left(c-4d\right)}\\ =-\frac{3}{cd}\end{array}$

Hence the simplified form of the expression is $-\frac{3}{cd}$ .