Exponentiation of Negative Numbers

Negative number raised to an even power

Raising any negative number to an even power will result in a positive outcome.
When nn is even:
(x)n=xn(-x)^n=x^n

Negative number raised to an odd power

Raising any negative number to an odd power will result in a negative outcome.
When nn is odd:
(x)n=(x)n(-x)^n=-(x)^n

What is the difference between a power that is inside parentheses and one that is outside of them?

When the exponent is outside the parentheses - it applies to everything inside them.
When the exponent is inside the parentheses - it applies only to its base and not to the minus sign that precedes it.

Practice Powers of Negative Numbers

Examples with solutions for Powers of Negative Numbers

Exercise #1

(8)2= (-8)^2=

Video Solution

Step-by-Step Solution

When we have a negative number raised to a power, the location of the minus sign is very important.

If the minus sign is inside or outside the parentheses, the result of the exercise can be completely different.

 

When the minus sign is inside the parentheses, our exercise will look like this:

(-8)*(-8)=

Since we know that minus times minus is actually plus, the result will be positive:

(-8)*(-8)=64

 

Answer

64 64

Exercise #2

(5)3=? (-5)^{-3}=\text{?}

Video Solution

Step-by-Step Solution

First let's recall the negative exponent rule:

bn=1bn b^{-n}=\frac{1}{b^n} We'll apply it to the expression we received:

(5)3=1(5)3 (-5)^{-3}=\frac{1}{(-5)^3} Next let's recall the power rule for expressions in parentheses:

(xy)n=xnyn (x\cdot y)^n=x^n\cdot y^n And we'll apply it to the denominator of the expression we received:

1(5)3=1(15)3=1(1)353=1153=153=1125 \frac{1}{(-5)^3}=\frac{1}{(-1\cdot5)^3}=\frac{1}{(-1)^3\cdot5^3}=\frac{1}{-1\cdot5^3}=-\frac{1}{5^3}=-\frac{1}{125} In the first step, we expressed the negative number inside the parentheses in the denominator as a multiplication between a positive number and negative one, and then we used the power rule for expressions in parentheses to expand the parentheses, and then we simplified the expression.

Let's summarize the solution to the problem:

(5)3=1(5)3=153=1125 (-5)^{-3}=\frac{1}{(-5)^3} =\frac{1}{-5^3}=-\frac{1}{125}

Therefore, the correct answer is answer B.

Answer

1125 -\frac{1}{125}

Exercise #3

(2)7= (-2)^7=

Video Solution

Answer

128 -128

Exercise #4

9= 9=

Video Solution

Answer

(3)2 (-3)^2

Exercise #5

(2)2= -(2)^2=

Video Solution

Answer

4 -4

Exercise #6

(1)80= -(-1)^{80}=

Video Solution

Answer

1 -1

Exercise #7

36= 36=

Video Solution

Answer

(6)2 (-6)^2

Exercise #8

49= 49=

Video Solution

Answer

(7)2 (-7)^2

Exercise #9

62= -6^2=

Video Solution

Answer

36 -36

Exercise #10

64= 64=

Video Solution

Answer

(8)2 (-8)^2

Exercise #11

8= 8=

Video Solution

Answer

(2)3 -(-2)^3

Exercise #12

(1)100= -(-1)^{100}=

Video Solution

Answer

1 -1

Exercise #13

(1)99= (-1)^{99}=

Video Solution

Answer

1 -1

Exercise #14

(2)3= -(-2)^3=

Video Solution

Answer

8 8

Exercise #15

(6)2= -(-6)^2=

Video Solution

Answer

36 -36