﻿ Equal or Opposites?
EQUAL OR OPPOSITES?
by Dr. Carol JVF Burns (website creator)
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• PRACTICE (online exercises and printable worksheets)
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Raising a Number and its Opposite to the same Even Power

When you raise a number and its opposite to the same even power,
then you get the same result.
That is:

For all real numbers $\,x\,$, $\,(-x)^{\text{EVEN}} = x^{\text{EVEN}}\,$.

Why?
Since $\,-1\,$ to any even power is $\,1\,$, we have: $$\cssId{s11}{(-x)^{\text{EVEN}}} \ \cssId{s12}{=\ (-1\cdot x)^{\text{EVEN}}} \ \cssId{s13}{=\ (-1)^{\text{EVEN}}x^{\text{EVEN}}} \ \cssId{s14}{=\ 1\cdot x^{\text{EVEN}}} \ \cssId{s15}{=\ x^{\text{EVEN}}}$$

Raising a Number and its Opposite to the same Odd Power

When you raise a number and its opposite to the same odd power,
then you get opposites as the result.
That is:

For all real numbers $\,x\,$, $\,(-x)^{\text{ODD}} = -x^{\text{ODD}}\,$.

Make sure you understand what this last mathematical sentence is saying:

$$\overset{\cssId{s23}{\text{this}}}{\overbrace{\cssId{s22}{(-x)^{\text{ODD}}}}} \quad \overset{\cssId{s25}{\text{is}}}{\overbrace{\cssId{s24}{\quad\quad=\strut\quad\quad}}} \quad \overset{\cssId{s27}{\text{the opposite of}}}{\overbrace{\cssId{s26}{\quad\quad-\strut\quad\quad}}} \quad \overset{\cssId{s29}{\text{this}}}{\overbrace{\cssId{s28}{x^{\text{ODD}}}}}$$

Why?
Since $\,-1\,$ to any odd power is $\,-1\,$, we have: $$\cssId{s33}{(-x)^{\text{ODD}}} \ \cssId{s34}{=\ (-1\cdot x)^{\text{ODD}}} \ \cssId{s35}{=\ (-1)^{\text{ODD}}x^{\text{ODD}}} \ \cssId{s36}{=\ -1\cdot x^{\text{ODD}}} \ \cssId{s37}{=\ -x^{\text{ODD}}}$$

EXAMPLES:

Determine if the expressions are EQUAL or OPPOSITES.

$(-x)^2\,$ and $\,x^2\,$ are equal
$(-x)^3\,$ and $\,x^3\,$ are opposites
$(-x)^4\,$ and $\,-x^4\,$ are opposites
$(-x)^5\,$ and $\,-x^5\,$ are equal
Master the ideas from this section
Recognizing the Patterns $\,x^n\,$ and $\,(-x)^n\,$