Equal or Opposites?

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 $\,(-x)^n\,$

(an even number, please)