Math Lesson 1.4.8 - Powers of Negative Numbers

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Welcome to our Math lesson on Powers of Negative Numbers, this is the eighth lesson of our suite of math lessons covering the topic of Order of Operations and PEMDAS Rule, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Powers of Negative Numbers

As we have explained earlier, powers are repeated multiplications of the same factor. When a negative number is raised in a certain power we apply the same rules as in multiplication. Thus, if we have a negative number raised in an even power, the result is positive as we can do the operations two by two.

On the other hand, when a negative number is raised by an odd power, the result is negative, as when exponents are grouped two by two they give a positive number, which, when multiplied with the only negative number left, becomes negative.

Let's consider a few examples in this regard:

Example 9

Calculate:

(-6)4 =
(-4)5 =
(-1)1027 =

Solution 9

We can write
(-6)4 = (-6) × (-6) × (-6) × (-6)
= [(-6) × (-6)] × [(-6) × (-6)]
= (+36) × (+36)
= (+1296)
We can write
(-4)5 = (-4) × (-4) × (-4) × (-4) × (-4)
= [(-4) × (-4)] × [(-4) × (-4)] × (-4)
= (+16) × (+16) × (-4)
= (+256) × (-4)
= (-1024)
In this case, we cannot write all factors, so we can write
(-1)1027 = (-1)1026 × (-1)
= (+1) × (-1)
= (-1)

Remarks:

Here, we have applied the property (-1)odd = (-1) & (-1)even = (+1).
Every number raised by the power 0 is equal to 1. We will explain this property in more detail in Chapter 7.

More Order of Operations and PEMDAS Rule Lessons and Learning Resources

Arithmetic Learning Material
Tutorial ID	Math Tutorial Title	Revision Notes	Revision Questions
1.4	Order of Operations and PEMDAS Rule
Lesson ID	Math Lesson Title	Lesson	Video Lesson
1.4.1	Order of Operation in Arithmetic Expressions Containing Only Addition and Subtraction
1.4.2	Order of Operation in Arithmetic Expressions Containing Only Multiplications and Division
1.4.3	Expressions Containing Additions, Subtractions, Multiplications and Divisions
1.4.4	Expressions Containing Exponents
1.4.5	Expressions Containing Brackets. PEMDAS Rule
1.4.6	Types of Parenthesis and Their Order of Priority
1.4.7	Working with Negative Numbers
1.4.8	Powers of Negative Numbers

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