Chapter 5.7, Problem 5ST

To determine

To simplify the given expression.

Answer to Problem 5ST

  $\frac{a^2-2ab+b^2}{a^2{b}^2}$

Explanation of Solution

Given:

Expression: $\frac{1}{a^2}-\frac{2}{ab}+\frac{1}{b^2}$

Concept used:

To simplify the given expression:

1. Find the Least Common Multiple of the fraction and write in reduced form.
2. Take out common terms in the given expression.
3. Cancel out the terms that are in common in numerator and the denominator.
4. Find if any algebraic identities can simplify the given expression.

Calculation:

Using the above concept,

  $\Rightarrow \frac{1}{a^2}-\frac{2}{ab}+\frac{1}{b^2}$

Least Common Multiple of $(a^2,ab,b^2)$ is $a^2{b}^2.$

  $\begin{array}{l}\Rightarrow \frac{1\times b^2}{a^2\times b^2}-\frac{2\times ab}{ab\times ab}+\frac{1\times a^2}{b^2\times a^2}\\ \Rightarrow \frac{b^2}{a^2{b}^2}-\frac{2ab}{a^2{b}^2}+\frac{a^2}{a^2{b}^2}\\ \Rightarrow \frac{b^2-2ab+a^2}{a^2{b}^2}\\ \Rightarrow \frac{a^2-2ab+b^2}{a^2{b}^2}\end{array}$

Conclusion:

Therefore, the simplified form of the expression is $\frac{a^2-2ab+b^2}{a^2{b}^2}.$