So far an exponent has counted factors, so it has been a positive whole number. What could $5^{0}$ or $2^{-3}$ mean? You can’t multiply something “zero times” or “minus three times.” We choose the meaning that keeps the pattern of powers and the quotient rule working.
Why it works
Going down the powers of 2, each step lowers the exponent by 1 and divides by 2: $2^{3} = 8$, $2^{2} = 4$, $2^{1} = 2$. One more step gives $2^{0} = 2 \div 2 = 1$. Keep going: $2^{-1} = \frac{1}{2}$, $2^{-2} = \frac{1}{4}$, $2^{-3} = \frac{1}{8}$.
The quotient rule agrees. $\frac{x^{3}}{x^{3}} = x^{3-3} = x^{0}$, and anything (not 0) divided by itself is 1. In $\frac{x^{2}}{x^{5}}$, three factors are left on the bottom, so $x^{2-5} = x^{-3}$ must equal $\frac{1}{x^{3}}$.
The quotient rule agrees. $\frac{x^{3}}{x^{3}} = x^{3-3} = x^{0}$, and anything (not 0) divided by itself is 1. In $\frac{x^{2}}{x^{5}}$, three factors are left on the bottom, so $x^{2-5} = x^{-3}$ must equal $\frac{1}{x^{3}}$.
Key idea
For any $a \ne 0$: $$a^{0} = 1 \qquad a^{-n} = \frac{1}{a^{n}}$$ A negative exponent means “divide by the base that many times.” It gives a small positive number, not a negative one.
Worked examples
Worked example: A negative exponent
Find the value of $4^{-2}$.
- $4^{-2} = \dfrac{1}{4^{2}}$
- $4^{2} = 16$, so $4^{-2} = \frac{1}{16}$.
Answer: $\frac{1}{16}$
Worked example: A fraction to a negative power
Find the value of $\left(\frac{2}{3}\right)^{-2}$.
- $\left(\frac{2}{3}\right)^{-2} = \dfrac{1}{(2/3)^{2}} = \dfrac{1}{4/9}$.
- Dividing by a fraction is multiplying by its reciprocal: $1 \cdot \frac{9}{4}$.
- Shortcut: a negative exponent flips the fraction, $\left(\frac{3}{2}\right)^{2} = \frac{9}{4}$.
Answer: $\frac{9}{4}$
Worked example: Watch the parentheses
Find $(-3)^{0}$ and $-3^{0}$.
- In $(-3)^{0}$ the base is $-3$, so the value is 1.
- In $-3^{0}$ the exponent belongs to the 3 only: $-3^{0} = -(3^{0}) = -1$.
Answer: $1$ and $-1$
Common mistake
Reading $2^{-3}$ as $-8$, as if the minus sign made the number negative. Check with the pattern: $2^{0} = 1$, and each step down halves, so $2^{-3}$ must be between 0 and 1. It is $\frac{1}{8}$.
Zero is left out: $0^{0}$ and negative powers of 0 have no value, because they would mean dividing by 0.