Math Core

How would you multiply $2^{10}$ by $2^{20}$ ? Writing out thirty 2s is no fun. Luckily, every exponent counts factors, and that one fact gives you a handful of exponent rules that turn long multiplications into quick addition and subtraction.

Multiplying powers with the same base

Look at $x^3 \cdot x^4$ written out:

x^3 \cdot x^4 = (x \cdot x \cdot x)(x \cdot x \cdot x \cdot x) = x^7

Three factors plus four factors makes seven factors. So when you multiply powers with the same base, keep the base and add the exponents.

Dividing powers with the same base

Now look at $\dfrac{6^5}{6^2}$ . Each factor of 6 on the bottom cancels one on top, since $\dfrac{6}{6} = 1$ :

\frac{6^5}{6^2} = \frac{6 \cdot 6 \cdot 6 \cdot \cancel{6} \cdot \cancel{6}}{\cancel{6} \cdot \cancel{6}} = 6^3

Five factors minus the two that cancel leaves three. When you divide powers with the same base, keep the base and subtract the exponents (top minus bottom).

Powers of powers and powers of products

In $(y^2)^3$ , the outside exponent says to use $y^2$ as a factor three times:

(y^2)^3 = y^2 \cdot y^2 \cdot y^2 = y^{2+2+2} = y^6

Three groups of 2 factors is $2 \cdot 3 = 6$ factors, so you multiply the exponents.

When the base is a product, like $(5a)^2$ , the exponent applies to each factor inside: $(5a)^2 = (5a)(5a) = 5 \cdot 5 \cdot a \cdot a = 25a^2$ .

Exponent rules

When the bases are the same and $a$ and $b$ are not zero:

Rule	In symbols	Example
Multiplying: add exponents	$a^m \cdot a^n = a^{m+n}$	$5^2 \cdot 5^6 = 5^8$
Dividing: subtract exponents	$\dfrac{a^m}{a^n} = a^{m-n}$	$\dfrac{n^9}{n^4} = n^5$
Power of a power: multiply exponents	$(a^m)^n = a^{m \cdot n}$	$(3^4)^2 = 3^8$
Power of a product	$(ab)^n = a^n b^n$	$(2y)^3 = 8y^3$

Worked example: Writing as a single power

Write each expression as a single power.

$9^4 \cdot 9^7$
$\dfrac{k^{15}}{k^5}$
$(2^5)^3$
$m \cdot m^6$

Solutions.

Same base, multiplying: $9^{4+7} = 9^{11}$ .
Same base, dividing: $k^{15-5} = k^{10}$ .
Power of a power: $2^{5 \cdot 3} = 2^{15}$ .
$m$ is $m^1$ , so $m^1 \cdot m^6 = m^{1+6} = m^7$ .

Common mistake

These rules only work when the bases match. You can't combine $4^3 \cdot 7^2$ into one power. And the base never changes: $2^3 \cdot 2^4 = 2^7$ , not $4^7$ . You are counting factors of 2, not multiplying the 2s together.

Zero as an exponent

What happens if you divide a power by itself? Try $\dfrac{4^3}{4^3}$ two ways:

Any nonzero number divided by itself is 1, so $\dfrac{4^3}{4^3} = 1$ .
Subtracting exponents gives $4^{3-3} = 4^0$ .

Both answers must be equal, so $4^0 = 1$ . The same reasoning works for any nonzero base.

Negative exponents

Keep dividing and the exponent can go below zero. Look at $\dfrac{10^2}{10^5}$ :

\frac{10^2}{10^5} = \frac{\cancel{10} \cdot \cancel{10}}{\cancel{10} \cdot \cancel{10} \cdot 10 \cdot 10 \cdot 10} = \frac{1}{10^3}

The subtraction rule says the answer is $10^{2-5} = 10^{-3}$ . So $10^{-3}$ means $\dfrac{1}{10^3} = \dfrac{1}{1{,}000} = 0.001$ . A negative exponent does not make a number negative. It tells you to take the reciprocal.

Zero and negative exponents

For any nonzero number $a$ :

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

Here is the pattern for powers of 10. Each step to the right divides by 10:

$10^3$	$10^2$	$10^1$	$10^0$	$10^{-1}$	$10^{-2}$	$10^{-3}$
$1{,}000$	$100$	$10$	$1$	$0.1$	$0.01$	$0.001$

Worked example: Zero and negative exponents

Evaluate each expression.

$15^0$
$2^{-3}$
$\dfrac{3^4}{3^6}$

Solutions.

Any nonzero number to the zero power is 1: $15^0 = 1$ .
$2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$ .
Subtract exponents: $3^{4-6} = 3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}$ .

Worked example: Combining rules

Write $\dfrac{(x^2)^5 \cdot x^3}{x^4}$ as a single power of $x$ .

Work from the inside out:

\begin{aligned} \frac{(x^2)^5 \cdot x^3}{x^4} &= \frac{x^{10} \cdot x^3}{x^4} && 2 \cdot 5 = 10 \\ &= \frac{x^{13}}{x^4} && 10 + 3 = 13 \\ &= x^9 && 13 - 4 = 9 \end{aligned}

Tip

Not sure which rule to use? Write out a tiny case, like $a^2 \cdot a^3$ , and count the factors. The rules are just shortcuts for counting.