Zero and Negative Exponents

The Zero Exponent

What is $2^0$? Using the division law:

But any number divided by itself is 1.

Therefore:

Negative Exponents

What happens if we continue dividing? Start with $2^3 = 8$.

• Divide by 2 $\rightarrow 2^2 = 4$
• Divide by 2 $\rightarrow 2^1 = 2$
• Divide by 2 $\rightarrow 2^0 = 1$
• Divide by 2 $\rightarrow 2^{-1} = \frac{1}{2}$
• Divide by 2 $\rightarrow 2^{-2} = \frac{1}{4} = \frac{1}{2^2}$

General Rule:

Visualizing on a Number Line

We can map powers on a vertical line. Notice how positive powers grow large upwards, while negative powers get closer and closer to zero (but never touch it!).