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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 \( \frac{x - y + 1}{{(x - y)^2}} \).

Step by step solution

01

Rewrite The Expressions

For this exercise, it might be easier to understand if we rewrite the exponents as fractions. So, we get \( \frac{1}{{x - y}} \) for the first part and \( \frac{1}{{(x - y)^2}} \) for the second part of the expression.
02

Find Common Denominator

To add these fractions, you need to have a common denominator. The common denominator for these two fractions is \( (x - y)^2 \). Therefore, rewrite the first fraction with the new denominator and it becomes \( \frac{x - y}{{(x - y)^2}} \).
03

Add the Fractions

Now we can add the two fractions together. This can be computed as \( \frac{x - y}{{(x - y)^2}} + \frac{1}{{(x - y)^2}} \).
04

Simplify the Answer

By adding them together we get \( \frac{x - y + 1}{{(x - y)^2}} \) which is our final answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When dealing with fractions, having a common denominator is crucial for addition and subtraction. You must ensure that the denominators (the bottom part of a fraction) are the same. Without this, you cannot directly add or subtract the fractions.
Most of the time, finding a common denominator involves finding the least common multiple (LCM) of the denominators. But sometimes, especially with algebraic fractions, it might be more about multiplying known terms.
  • For instance, with the expressions \( \frac{1}{x-y} \) and \( \frac{1}{(x-y)^2} \), the denominators are \((x-y)\) and \((x-y)^2\).
  • The smallest expression that both \((x-y)\) and \((x-y)^2\) can divide into evenly is \((x-y)^2\), making it the common denominator.
By using a common denominator, you can transform fractions, ensuring they are easily combinable.
Exponents
Exponents are used to represent repeated multiplication of a base number. They are a powerful mathematical notation.
When dealing with negative exponents, like \( (x-y)^{-1} \), it signifies the reciprocal. In other words, it's the same as saying \( \frac{1}{x-y} \).
  • Similarly, \( (x-y)^{-2} \) represents \( \frac{1}{(x-y)^2} \).
Understanding exponents can simplify complex expressions by allowing you to express powers and roots efficiently.
Simplifying Fractions
Simplifying fractions involves reducing a fraction to its simplest form, where both the numerator and the denominator have no common factors other than 1.
In the case of algebraic fractions, simplifying requires careful manipulation of the terms. It might involve factoring, canceling out common terms, or reorganizing the expression.
For instance:
  • When you add \( \frac{x-y}{(x-y)^2} + \frac{1}{(x-y)^2} \), you combine numerators over a shared denominator: \( \frac{x-y+1}{(x-y)^2} \).
  • Check if the numerator and the denominator can be further simplified by factoring common elements or expressions.
Simplification makes fractions easier to interpret and work with, especially in larger and more complex calculations.

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Most popular questions from this chapter

The formula for converting Celsius temperature, \(C,\) to Fahrenheit temperature, \(F\), is $$F=\frac{9}{5} C+32$$ If Fahrenheit temperature ranges from \(41^{\circ}\) to \(50^{\circ},\) inclusive, what is the range for Celsius temperature? Use interval notation to express this range.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\)

Will help you prepare for the material covered in the next section. A telephone texting plan has a monthly fee of 20 dollar with a charge of 0.05 dollar per text. Write an algebraic expression that models the plan's monthly cost for \(x\) text messages.

Solve each equation. $$7-7 x=(3 x+2)(x-1)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, \(x,\) is modeled by \(|x-20| \leq 5\)
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