Equal or Opposites?
EQUAL OR OPPOSITES?
• PRACTICE (online exercises and printable worksheets)
• This page gives an in-a-nutshell discussion of the concepts.
Want more details, more exercises? Read the full text!

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: $$(-x)^{\text{EVEN}} \ =\ (-1\cdot x)^{\text{EVEN}} \ =\ (-1)^{\text{EVEN}}x^{\text{EVEN}} \ =\ 1\cdot x^{\text{EVEN}} \ =\ x^{\text{EVEN}}$$

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: $$\overset{\text{this}}{\overbrace{(-x)^{\text{ODD}}}} \quad \overset{\text{is}}{\overbrace{\quad\quad=\strut\quad\quad}} \quad \overset{\text{the opposite of}}{\overbrace{\quad\quad-\strut\quad\quad}} \quad \overset{\text{this}}{\overbrace{x^{\text{ODD}}}}$$

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

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
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Recognizing the Patterns $\,x^n\,$ and $\,(-x)^n\,$

Equal or Opposites?

 (an even number, please)