Math Lesson 7.1.4 - Powers of Negative Numbers

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

Powers of Negative Numbers

In this paragraph, we discuss the sign of the result obtained by raising a number at a given power. Thus, we don't have concerns if a positive number (i.e. if the base is positive) is raised at a given power because the result is always positive regardless of the sign of the index. Thus, if the index is positive, it is obvious that we always obtain a positive number as a result because in such situations we multiply a positive number several times (determined by the value of index) by itself.

However, if a negative number is raised at a given power, we must be very careful about the sign of the final result because from the table of sign rules provided in tutorial 6.2, it is clear that:

If a negative number is raised at an even power (the index is an even number), the result is positive and when a negative number is raised at an odd power (the index is an odd number) the result is negative.

For example,

(-2)³ = -8

because

(-2)³ = (-2) ∙ (-2) ∙ (-2)
= [(-2) ∙ (-2)] ∙ (-2)
= ( + 4) ∙ (-2)
= -8

On the other hand,

(-2)⁴ = + 16

because

(-2)⁴ = (-2) ∙ (-2) ∙ (-2) ∙ (-2)
= [(-2) ∙ (-2)] ∙ [(-2) ∙ (-2)]
= ( + 4) ∙ ( + 4)
= + 16

Example 4

Calculate the value of the following expressions:

(- 3)⁴/(-2)³
(- 1)¹⁶⁷ · (- 2)³ · (- 5)²

Solution 4

We have
(-3)⁴/(-2)³ = 81/-8 = -81/8
We have
(-1)¹⁶⁷ ∙ (-2)³ ∙ (-5)²
= (-1) ∙ (-8) ∙ ( + 2)
= [(-1) ∙ (-8)] ∙ ( + 2)
= ( + 8) ∙ ( + 2)
= + 16

Putting All Together

Now, let's see a couple of examples where all the above properties of indices can be applied.

Example 5

Simplify the value of the following algebraic expressions.

3^{2 + x} + 3^x/2^x ∙ 5^x
(2⁴ ∙ 4^-2/3⁵ ∙ 9^-3)^-2

Solution 5

We have:
3^{2 + x} + 3^x/2^x ∙ 5^x = 3² ∙ 3^x + 3^x/(2 ∙ 5)^x
= 3^x (3² + 1)/(2 ∙ 5)^x
= 3^x ∙ 10/10^x
= 3^x ∙ 10¹/10^x
= 3^x ∙ 10^{1 - x}
We have
(2⁴ ∙ 4^-2/3⁵ ∙ 9^-3)^-2 = [2⁴ ∙ (2² )^-2/3⁵ ∙ (3² )^-3 ]^-2
= [2⁴ ∙ 2^{2 ∙ (-2)}/3⁵ ∙ 3^{2 ∙ (-3)}]^-2
= [2⁴ ∙ 2^-4/3⁵ ∙ 3^-6]^-2
= [2^{4 + (-4)}/3^{5 + (-6)} ]^-2
= [ 2⁰/3^-1 ]^-2
= [3^-1/2⁰ ]²
= (1/1 ∙ 3)²
= (1/3)²
= 1²/3²
= 1/9

More Indices Lessons and Learning Resources

Powers and Roots Learning Material
Tutorial ID	Math Tutorial Title	Revision Notes	Revision Questions
7.1	Indices
Lesson ID	Math Lesson Title	Lesson	Video Lesson
7.1.1	Powers, Indices, and Exponents
7.1.2	Properties of Indices
7.1.3	Negative Indices. The Meaning of Reciprocal
7.1.4	Powers of Negative Numbers

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