Chapter 6.5, Problem 46WE

To determine

Find the simplified form of the given expression.

Answer to Problem 46WE

The simplified form of the expression is $\frac{4}{\left(n-5\right)\left(n+5\right)}$ .

Explanation of Solution

Given:

The given expression is $\frac{2n}{n^3-5n^2}+\frac{2}{n^2+5n}$ .

Calculation:

Rewrite the given expression.

  $\begin{array}{l}\frac{2n}{n^3-5n^2}+\frac{2}{n^2+5n}\\ =\frac{2n}{n^2\left(n-5\right)}+\frac{2}{n\left(n+5\right)}\end{array}$

Since the denominators are different, rewrite the fractions using their least common denominator, $\left(x+1\right)\left(x-1\right)$ .

  $\begin{array}{l}=\frac{2n\left(n+5\right)}{n^2\left(n-5\right)\left(n+5\right)}+\frac{2n\left(n-5\right)}{n\left(n+5\right)n\left(n-5\right)}\\ =\frac{2n\left(n+5\right)}{n^2\left(n-5\right)\left(n+5\right)}+\frac{2n\left(n-5\right)}{n^2\left(n-5\right)\left(n+5\right)}\end{array}$

Useaddition rule of fraction.

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

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

  $\begin{array}{l}=\frac{2n^2+10n+2n^2-10n}{n^2\left(n-5\right)\left(n+5\right)}\\ =\frac{4n^2}{n^2\left(n-5\right)\left(n+5\right)}\\ =\frac{4}{\left(n-5\right)\left(n+5\right)}\end{array}$

Hence the simplified form of the expression is $\frac{4}{\left(n-5\right)\left(n+5\right)}$ .