Does (x + y)^(-2) = x^(-2) + y^(-2)?

Question:

Does {eq}(x + y)^{-2} = x^{-2} + y^{-2} {/eq}?

Expanding

Here we have to show that when we expand the terms inside the bracket on the left-hand side of the equation, the result is equal to the right-hand side. This can be done by isolating the left-hand-side and working on it until it is equal to the other side.

Answer and Explanation:


Let's expand the term on the left-hand side of the equation.


$$\begin{align} (x + y)^{-2}&=\frac{1}{(x + y)^2}&&&&\because a^{-b}=\frac{1}{a^b}\text{ with (x+y)=a }\\ &=\frac{1}{x^2+y^2+2xy} \end{align} $$


As can be seen, the two sides of the equation are not the same. So, we can say that {eq}(x + y)^{-2}\neq x^{-2} + y^{-2} {/eq}.


Learn more about this topic:

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Evaluating Simple Algebraic Expressions

from ELM: CSU Math Study Guide

Chapter 6 / Lesson 3
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