Chapter 6.6, Problem 31WE

To determine

Simplify: the given expression as fraction.

Answer to Problem 31WE

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

Explanation of Solution

Given information:

The expression $\left(a+\frac{2}{a}\right)\left(a-\frac{2}{a}\right)$.

Formula used:

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

  $\left(a+b\right)\left(a-b\right)=a^2-b^2$

  $\left(a+\frac{2}{a}\right)\left(a-\frac{2}{a}\right)$

Factorizing the expression using identity $\left(a+\frac{2}{a}\right)\left(a-\frac{2}{a}\right)$.

  $\begin{array}{l}=a^2-{\left(\frac{2}{a}\right)}^2\\ =a^2-\frac{4}{a^2}\end{array}$

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

  $=\frac{a^2}{1}-\frac{4}{a^2}$

Least common denominator for $a^2,\frac{4}{a^2}$ is $a^2$ . So,

  $\begin{array}{l}=\frac{a^2\cdot a^2-4}{a^2}\\ =\frac{a^4-4}{a^2}\\ =\frac{{\left(a^2\right)}^2-2^2}{a^2}\end{array}$

Factorizing the numerator of expression by using identity $a^2-b^2=\left(a+b\right)\left(a-b\right)$.

  $=\frac{\left(a^2-2\right)\left(a^2+2\right)}{a^2}$

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