Chapter 5.4, Problem 7WE

To determine

To simplify the given expression.

Answer to Problem 7WE

  $\frac{s+t}{s-t}$

Explanation of Solution

Given:

Expression: $\frac{s^2-t^2}{{(t-s)}^2}$

Concept used:

To simplify the given expression:

1. Take out common terms in the given expression.
2. Cancel out the terms that are in common in numerator and the denominator.
3. Find if any algebraic identities can simplify the given expression.

Formula used:

  $(a^2-b^2)=(a-b)(a+b)$

Calculation:

In the given expression find out the common term in the numerator and denominator.

  $\begin{array}{l}\Rightarrow \frac{s^2-t^2}{{(t-s)}^2}\\ \Rightarrow \frac{(s-t)(s+t)}{{(t-s)}^2}[\because (a^2-b^2)=(a-b)(a+b)]\\ $ \Rightarrow \frac{(s-t)(s+t)}{(t-s)(t-s)}[\because {(a-b)}^2=(a-b)(a-b)]$\Rightarrow \frac{\cancel{(s-t)}(s+t)}{-\cancel{(s-t)}(t-s)}[\because (s-t)=-(t-s)]\\ \Rightarrow \frac{(s+t)}{-(t-s)}\end{array}$

  $\begin{array}{l}\Rightarrow \frac{s+t}{-t+s}\\ \Rightarrow \frac{s+t}{s-t}\end{array}$

Conclusion:

Therefore, the simplified expression is $\frac{s+t}{s-t}.$