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Chapter 0: Problem 119

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

Short Answer

Expert verified
The simplified form of the expression is \((x + 1 - y) / (x - y)^2\)

Step by step solution

01

Rewrite the expressions

Convert the expressions with negative exponents into expressions with positive exponents by taking the reciprocal. The reciprocal of a term is simply one divided by that term. Substituting \(-1\) by \(1/(x - y)\) and \(-2\) by \(1/(x - y)^2\), the expression becomes \(1/(x - y) + 1/(x - y)^2\).
02

Find common denominator

Notice that the two terms do not have the same denominator - they cannot be directly added. First, you'll need to find a common denominator. For this, notice that you can express \(1/(x - y)\) as \((x - y)/(x - y)^2\). Now, both terms have a common denominator of \((x - y)^2\). So, the expression is now \((x - y)/(x - y)^2 + 1/(x - y)^2\).
03

Combine like terms

Now that they have the same denominator, you can combine the fractions. You add the numerators together and keep the denominator the same. So, the expression simplifies to \((1 + x - y)/(x - y)^2\).
04

Final simplification

Group terms in the numerator that are alike so it's easier to see what the expression looks like. So, the final simplified form of the expression is \((x + 1 - y) / (x - y)^2\)

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \%, 88 \%, 92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

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