Chapter 6.6, Problem 17WE

To determine

Simplify: the given expression as fraction.

Answer to Problem 17WE

The simplified form of $8n+\frac{2}{n+1}$ is $\frac{2{\left(2n+1\right)}^2}{n+1}$.

Explanation of Solution

Given information:

The expression $8n+\frac{2}{n+1}$.

Formula used:

Least common denominator: It is the smallest multiple of denominator of set of fractions.

  $a^2+2ab+b^2={\left(a+b\right)}^2$

  $8n+\frac{2}{n+1}$

Recall that Least common denominator is the smallest multiple of denominator of set of fractions.

  $=\frac{8n}{1}+\frac{2}{n+1}$

Least common denominator for $\frac{8n}{1},\frac{2}{n+1}$ is $n+1$ . So,

  $\begin{array}{l}=\frac{8n\left(n+1\right)+2}{n+1}\\ =\frac{8n^2+8n+2}{n+1}\\ =\frac{2\left(4n^2+4n+1\right)}{n+1}\\ =\frac{2\left({\left(2n\right)}^2+2\cdot 2n+1^2\right)}{n+1}\end{array}$

  $=\frac{2{\left(2n+1\right)}^2}{n+1}$

The simplified form of $8n+\frac{2}{n+1}$ is $\frac{2{\left(2n+1\right)}^2}{n+1}$.