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
[beautiful math coming... please be patient]
$\,x\,$,
[beautiful math coming... please be patient]
$\,(-x)^{\text{EVEN}} = x^{\text{EVEN}}\,$.
Why?
Since $\,-1\,$ to any even power is $\,1\,$, we have:
$$
(-x)^{\text{EVEN}} \ =\ (-1\cdot x)^{\text{EVEN}} \ =\ (-1)^{\text{EVEN}}x^{\text{EVEN}} \ =\ 1\cdot x^{\text{EVEN}} \ =\ x^{\text{EVEN}}
$$
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
[beautiful math coming... please be patient]
$\,x\,$,
[beautiful math coming... please be patient]
$\,(-x)^{\text{ODD}} = -x^{\text{ODD}}\,$.
Make sure you understand what this last mathematical sentence is saying: [beautiful math coming... please be patient] $$ \overset{\text{this}}{\overbrace{(-x)^{\text{ODD}}}} \quad \overset{\text{is}}{\overbrace{\quad\quad=\strut\quad\quad}} \quad \overset{\text{the opposite of}}{\overbrace{\quad\quad-\strut\quad\quad}} \quad \overset{\text{this}}{\overbrace{x^{\text{ODD}}}} $$
Why?
Since $\,-1\,$ to any odd power is $\,-1\,$, we have:
[beautiful math coming... please be patient]
$$
(-x)^{\text{ODD}} \ =\ (-1\cdot x)^{\text{ODD}} \ =\ (-1)^{\text{ODD}}x^{\text{ODD}} \ =\ -1\cdot x^{\text{ODD}} \ =\ -x^{\text{ODD}}
$$
Determine if the expressions are EQUAL or OPPOSITES.