Problem 108 - College Algebra

Question

Perform the indicated operations. $$(x-y)^{-1}+(x-y)^{-2}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{(x-y)+1}{(x-y)^2}\)

Step 1: Understanding the Negative Exponent Rule

In mathematics, a negative exponent refers to the reciprocal (or 'flip') of the base. Thus, \((x-y)^{-1}\) becomes \(\frac{1}{x-y}\), and \((x-y)^{-2}\) becomes \(\frac{1}{(x-y)^2}\). Therefore, the expression simplifies to: \(\frac{1}{x-y}+\frac{1}{(x-y)^2}\).

Step 2: Finding a Common Denominator

To add two fractions, a common denominator (same bottom number) is needed. The common denominator between \((x-y)\) and \((x-y)^2\) is \((x-y)^2\). Thus, the first fraction is multiplied by \((x-y)\) at the top and bottom. The expression then becomes: \(\frac{(x-y)}{(x-y)^2}+\frac{1}{(x-y)^2}\).

Step 3: Addition of fractions

Since the denominators of both fractions are the same, the fractions can now be added by adding the numerators and keeping the denominator constant. This gives the final simplified expression: \(\frac{(x-y)+1}{(x-y)^2}\).