Negative and Zero Exponents

You already know how to work with positive exponents, like $2^3 = 8$. But what happens when the exponent is zero or a negative number? By extending the rules of exponents, we can easily solve these problems.

The Zero Exponent Rule

The rule for zero exponents is simple: Any nonzero number raised to the zero power equals 1.

$a^0 = 1 \quad (a \neq 0)$

Why does this work? Think about the pattern of dividing by the base. Let's look at the powers of 2:

• $2^3 = 8$
• $2^2 = 4$ (divided 8 by 2)
• $2^1 = 2$ (divided 4 by 2)
• $2^0 = 1$ (divided 2 by 2)

Example: $5^0 = 1$

The Negative Exponent Rule

A negative exponent does not make the number negative! Instead, it means you take the reciprocal of the base and make the exponent positive.

$a^{-n} = \frac{1}{a^n} \quad (a \neq 0)$

If we continue our pattern from above by dividing by 2 again:

• $2^0 = 1$
• $2^{-1} = \frac{1}{2^1} = \frac{1}{2}$
• $2^{-2} = \frac{1}{2^2} = \frac{1}{4}$
• $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Negative Exponents with Fractions

When you have a fraction raised to a negative exponent, you can simply flip the fraction (find its reciprocal) and change the exponent to a positive number:

$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$

Example: $\left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2 = \frac{4^2}{3^2} = \frac{16}{9}$

Summary

• Zero Exponent: $x^0 = 1$
• Negative Exponent: $x^{-n} = \frac{1}{x^n}$
• Fraction with Negative Exponent: Flip the fraction and make the power positive.