Math Core

The quotient rule says $\dfrac{a^m}{a^n} = a^{m-n}$ . So what is $\dfrac{5^3}{5^3}$ ? The rule says $5^0$ , but you know any number divided by itself is 1. And $\dfrac{5^2}{5^6}$ gives $5^{-4}$ . What could a zero or negative exponent possibly mean? This lesson answers both questions, so the exponent rules work for every integer exponent.

Following the pattern

Look at the powers of 3, going down one exponent at a time:

Power	$3^4$	$3^3$	$3^2$	$3^1$	$3^0$	$3^{-1}$	$3^{-2}$
Value	$81$	$27$	$9$	$3$	?	?	?

Each time the exponent goes down by 1, the value is divided by 3. Keep the pattern going:

$3^0 = 3 \div 3 = 1$
$3^{-1} = 1 \div 3 = \dfrac{1}{3}$
$3^{-2} = \dfrac{1}{3} \div 3 = \dfrac{1}{9} = \dfrac{1}{3^2}$

The pattern works for any nonzero base, and it matches what the quotient rule told us: $\dfrac{5^3}{5^3} = 1$ , and

\frac{5^2}{5^6} = \frac{\cancel{5} \cdot \cancel{5}}{\cancel{5} \cdot \cancel{5} \cdot 5 \cdot 5 \cdot 5 \cdot 5} = \frac{1}{5^4}, \quad \text{so } 5^{-4} = \frac{1}{5^4}.

Zero and negative exponents

For any nonzero number $a$ and any whole number $n$ :

a^0 = 1 \qquad\qquad a^{-n} = \frac{1}{a^n}

A negative exponent means "take the reciprocal." It does not make the number negative.

( $0^0$ is left undefined, which is why the base must be nonzero.)

Worked example: Evaluating

Evaluate each expression.

$7^0$
$2^{-5}$
$10^{-3}$

Solutions.

Any nonzero number to the zero power is 1, so $7^0 = 1$ .
$2^{-5} = \dfrac{1}{2^5} = \dfrac{1}{32}$ .
$10^{-3} = \dfrac{1}{10^3} = \dfrac{1}{1{,}000} = 0.001$ .

Common mistake

A negative exponent never makes the answer negative. $4^{-2} = \dfrac{1}{16}$ , not $-16$ and not $-8$ .

Also watch the parentheses with zero exponents. In $(-6)^0$ the base is $-6$ , so $(-6)^0 = 1$ . In $-6^0$ the base is just $6$ , so $-6^0 = -(6^0) = -1$ .

Fractions with negative exponents

Taking the reciprocal of a fraction flips it over. So a negative exponent on a fraction flips the fraction and makes the exponent positive:

\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^{2} = \frac{25}{4}

The same idea lets you move a factor across the fraction bar: $\dfrac{1}{x^{-3}} = x^3$ .

The rules still work

All the properties from the last lesson hold for zero and negative exponents too. Add, subtract or multiply the exponents exactly as before, being careful with the signs.

Worked example: Using the properties with negative exponents

Simplify each expression. Write the answer as a single power, then evaluate.

$6^{-4} \cdot 6^{6}$
$\dfrac{2^{3}}{2^{7}}$
$(10^{-2})^{-3}$

Solutions.

Add: $-4 + 6 = 2$ . So $6^{-4} \cdot 6^6 = 6^2 = 36$ .
Subtract: $3 - 7 = -4$ . So $\dfrac{2^3}{2^7} = 2^{-4} = \dfrac{1}{16}$ .
Multiply: $(-2)(-3) = 6$ . So $(10^{-2})^{-3} = 10^6 = 1{,}000{,}000$ .

Worked example: Writing with positive exponents

Simplify $\dfrac{x^{-2} \cdot x^{5}}{x^{7}}$ and write the answer using only positive exponents.

\begin{aligned} \frac{x^{-2} \cdot x^{5}}{x^{7}} &= \frac{x^{3}}{x^{7}} && -2 + 5 = 3 \\ &= x^{-4} && 3 - 7 = -4 \\ &= \frac{1}{x^4} \end{aligned}

Tip

When you subtract exponents, write the subtraction out: $3 - (-2) = 5$ . Most sign mistakes happen when this step is done in your head.