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=1xnx^{-n}=\dfrac{1}{x^n}
Want to learn more about this definition? Check out this video.

Examples

  • 35=1353^{-5}=\dfrac{1}{3^5}
  • 128=28\dfrac{1}{2^8}=2^{-8}
  • y2=1y2y^{-2}=\dfrac{1}{y^{2}}
  • (86)3=(68)3\left(\dfrac{8}{6}\right)^{-3}=\left(\dfrac{6}{8}\right)^{3}

Practice

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

nn2n2^n
3323=82^3=8
2222=42^2=4
1121=22^1=2
0020=12^0=1
1-121=122^{-1}=\dfrac12
2-222=142^{-2}=\dfrac14
Notice how 2n2^n is divided by 22 each time we reduce nn. This pattern continues even when nn is zero or negative.

Justification #2: Exponent properties

Recall that xnxm=xnm\dfrac{x^n}{x^m}=x^{n-m}. So...
2223=223=21\begin{aligned} \dfrac{2^2}{2^3}&=2^{2-3} \\\\ &=2^{-1} \end{aligned}
We also know that
2223=22222=12\begin{aligned} \dfrac{2^2}{2^3}&=\dfrac{\cancel 2\cdot\cancel 2}{\cancel 2\cdot\cancel 2\cdot 2} \\\\ &=\dfrac12 \end{aligned}
And so we get 21=122^{-1}=\dfrac12.
Also, recall that xnxm=xn+mx^n\cdot x^m=x^{n+m}. So...
2222=22+(2)=20=1\begin{aligned} 2^2\cdot 2^{-2}&=2^{2+(-2)} \\\\ &=2^0 \\\\ &=1 \end{aligned}
And indeed, according to the definition...
2222=22122=2222=1\begin{aligned} 2^2\cdot 2^{-2}&=2^2\cdot\dfrac{1}{2^2} \\\\ &=\dfrac{2^2}{2^2} \\\\ &=1 \end{aligned}