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Negative exponents review

Review the basics of negative exponents and try some practice problems. 

Definition for negative exponents

We define a negative power as the multiplicative inverse of the base raised to the positive opposite of the power:
xn=1xn
Want to learn more about this definition? Check out this video.

Examples

  • 35=135
  • 128=28
  • y2=1y2
  • (86)3=(68)3

Practice

Problem 1
Select the equivalent expression.
43=?
Choose 1 answer:

Want to try more problems like these? Check out this exercise.

Some intuition

So why do we define negative exponents this way? Here are a couple of justifications:

Justification #1: Patterns

n2n
323=8
222=4
121=2
020=1
121=12
222=14
Notice how 2n is divided by 2 each time we reduce n. This pattern continues even when n is zero or negative.

Justification #2: Exponent properties

Recall that xnxm=xnm. So...
2223=223=21
We also know that
2223=22222=12
And so we get 21=12.
Also, recall that xnxm=xn+m. So...
2222=22+(2)=20=1
And indeed, according to the definition...
2222=22122=2222=1

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