audio read-through 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:

$$ \begin{align} \cssId{s11}{(-x)^{\text{EVEN}}}\ &\cssId{s12}{=\ (-1\cdot x)^{\text{EVEN}}}\cr\cr &\cssId{s13}{=\ (-1)^{\text{EVEN}}x^{\text{EVEN}}}\cr\cr &\cssId{s14}{=\ 1\cdot x^{\text{EVEN}}}\cr\cr &\cssId{s15}{=\ x^{\text{EVEN}}} \end{align} $$

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:

$$ \begin{align} \cssId{s33}{(-x)^{\text{ODD}}} \ &\cssId{s34}{=\ (-1\cdot x)^{\text{ODD}}}\cr\cr &\cssId{s35}{=\ (-1)^{\text{ODD}}x^{\text{ODD}}}\cr\cr &\cssId{s36}{=\ -1\cdot x^{\text{ODD}}}\cr\cr &\cssId{s37}{=\ -x^{\text{ODD}}} \end{align} $$

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

Practice

EQUAL
OPPOSITES