Exponents of Negative Numbers

In this article, you will learn everything you need to know about the exponentiation of negative numbers and understand the difference between a power that is inside the parentheses and another that appears outside of them.
Shall we start?

So far, we have learned to solve powers of positive numbers always obtaining positive results.
When a negative number is raised to a certain power, the result can be either positive or negative.

Raising any negative number to an exponent that is an even number, even power, will result in a positive outcome.

$(-3)^2=$
If we want to simplify the exercise we will get: $(-3) \times (-3)=$
Negative times negative = Positive
Therefore, the result will be $9$.
When the base is a negative number and the exponent is even, we can ignore the minus sign. Let's formulate it like this:
When $n$ is even:
$(-x)^n=x^n$

When raising any negative number to an exponent that is an odd number, odd power, the result will be negative.

$(-3)^3=$
If we want to simplify the exercise we will get: $(-3) \times (-3) \times (-3)=$
Negative times negative = Positive
Positive times negative = Negative
Therefore, the result will be $-27$.
When the base is a negative number and the exponent is odd, we cannot ignore the minus sign, the result will always be negative.
Let's formulate it as a rule:
When $n$ is odd:

$(-x)^n=-(x)^n$

$(-4)^3=$

Solution:
In this exercise, the exponent is odd.
Therefore, the result must necessarily be negative.
We will obtain:
$(-4) \times (-4) \times (-4)=-64$

$(-2)^4=$

Solution:
In this exercise, the exponent is even. Consequently, we can ignore the minus sign and the result will be positive.
We will obtain:
$(-2) \times (-2) \times (-2) \times (-2)=16$

$(-5)^5=$

Solution:
In this exercise, the exponent is odd. Consequently, the result will be negative.
We will obtain:
$(-5) \times (-5) \times (-5) \times (-5) \times (-5)=-3125$

It is important that you know that the difference is very large.

When the exponent appears outside of the parentheses
We multiply the number inside the parentheses by itself, as many times as indicated by the number representing the exponent.
For example:
$(-4)^2=$
$(-4) \times (-4)=16$

On the other hand, when the exponent is inside the parentheses (sometimes, without any parentheses)
Thus:
$(-4^2 )=$
or
$-(4^2 )=$
or
$-4^2=$
The exponent applies only and exclusively to the base number and not to the minus sign that precedes it.
Therefore, we will calculate the power and add the minus as an annex.
We will obtain:
$-4^2=-16$

We have obtained $2$ different answers! That's why it is necessary to pay close attention to understand well to which part of the exercise the power applies.
If the exponent is outside the parentheses - it applies to everything inside them.
If the exponent is inside the parentheses - it applies only to its base and not to the minus sign that precedes it.

Let's see how this is done in slightly more complicated exercises:

$(-2)^4-3^2=$

Solution:
Let's start with $(-2)^4$
The positive exponent is outside the parentheses, therefore, it applies to the entire $-2$.
We will obtain: $(-2)^4 = 16$
Let's rewrite the exercise, we will get:
$16-3^2=$
Now let's continue with the other part of the exercise.
In the second monomial, there are no parentheses, meaning the exponent applies only to the $3$ without taking into account the minus sign that precedes it.
We know that
$3^2= 9$
Therefore, let's rewrite the exercise in the following way:
$16-9=7$
Note –> Although it's true that the power is positive, it does not apply to the entire $-3$ therefore, we will not write $9$ but,$-9$.