This is one of the most common places students lose points on tests—simply because of a missing minus sign! But don’t worry, there is a simple pattern (an “Even vs. Odd” switch) that makes it easy to predict the answer.
Let’s start with the simplest negative number: $-1$.
Look at what happens when we multiply $-1$ by itself repeatedly:
$(-1)^2 = -1 \times -1 = \mathbf{1}$
$(-1)^3 = -1 \times -1 \times -1 = \mathbf{-1}$
$(-1)^4 = -1 \times -1 \times -1 \times -1 = \mathbf{1}$
The Pattern:
If the exponent is EVEN (2, 4, 6…), the negative signs cancel out in pairs, and the answer is Positive (+).
If the exponent is ODD (1, 3, 5…), one negative sign is left over, and the answer is Negative (-).
This rule applies to any negative number inside parentheses, not just $-1$.
If $n$ is EVEN: The result is positive ($a^n$).
If $n$ is ODD: The result is negative ($-a^n$).
Let’s look at the difference between an even and odd power using the number $-3$.
⚠️ Teacher’s Note:
Be very careful with parentheses!
$(-5)^2$ means “negative 5 squared,” which is positive 25.
$-5^2$ means “the negative of 5 squared,” which is -25.
The rule above only works when the negative sign is inside the parentheses!