Math Core

You already know that $2^5$ means five factors of 2 multiplied together. But what happens when you multiply $2^5$ by $2^3$ , or raise $2^5$ to the fourth power? Writing out every factor would work, but it gets long fast. A few simple properties of exponents let you simplify these expressions in one step.

Multiplying powers with the same base

Write out the factors of $3^4 \cdot 3^2$ :

3^4 \cdot 3^2 = (3 \cdot 3 \cdot 3 \cdot 3) \cdot (3 \cdot 3) = 3^6

There are 4 threes in the first group and 2 in the second, so there are $4 + 2 = 6$ threes in all. When the bases match, you just count the total number of factors by adding the exponents.

Dividing powers with the same base

Now try $\dfrac{5^6}{5^2}$ . Write out the factors and cancel pairs, because $\dfrac{5}{5} = 1$ :

\frac{5^6}{5^2} = \frac{5 \cdot 5 \cdot 5 \cdot 5 \cdot \cancel{5} \cdot \cancel{5}}{\cancel{5} \cdot \cancel{5}} = 5^4

Two factors on the bottom cancel two of the six on top, leaving $6 - 2 = 4$ . When you divide powers with the same base, subtract the exponents.

Raising a power to a power

What does $(4^3)^2$ mean? The exponent 2 says to use $4^3$ as a factor twice:

(4^3)^2 = 4^3 \cdot 4^3 = 4^{3 + 3} = 4^6

Two groups of 3 factors is $3 \cdot 2 = 6$ factors. So to raise a power to a power, multiply the exponents.

Powers of products and quotients

An exponent outside parentheses applies to every factor inside:

(2x)^3 = (2x)(2x)(2x) = (2 \cdot 2 \cdot 2)(x \cdot x \cdot x) = 2^3 x^3 = 8x^3

The same thing happens with a fraction: $\left(\dfrac{a}{b}\right)^2 = \dfrac{a}{b} \cdot \dfrac{a}{b} = \dfrac{a^2}{b^2}$ .

Properties of exponents

For any nonzero numbers $a$ and $b$ and positive whole numbers $m$ and $n$ :

Property	Rule	Example
Product of powers	$a^m \cdot a^n = a^{m+n}$	$x^4 \cdot x^3 = x^7$
Quotient of powers	$\dfrac{a^m}{a^n} = a^{m-n}$	$\dfrac{x^9}{x^2} = x^7$
Power of a power	$(a^m)^n = a^{mn}$	$(x^2)^5 = x^{10}$
Power of a product	$(ab)^n = a^n b^n$	$(3x)^2 = 9x^2$
Power of a quotient	$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$	$\left(\dfrac{x}{2}\right)^3 = \dfrac{x^3}{8}$

Every rule comes from the same idea: an exponent counts factors. If you ever forget a rule, write out a small example and count.

Worked example: Using one property at a time

Write each expression as a single power.

$7^5 \cdot 7^8$
$\dfrac{y^{12}}{y^4}$
$(10^3)^4$

Solutions.

Same base, multiplying: add the exponents. $7^{5+8} = 7^{13}$ .
Same base, dividing: subtract the exponents. $y^{12-4} = y^8$ .
Power of a power: multiply the exponents. $10^{3 \cdot 4} = 10^{12}$ .

Common mistake

The rules only combine exponents when the bases are the same. $2^3 \cdot 5^4$ cannot be written as one power, because the bases are different. Also, keep the base as it is: $3^4 \cdot 3^2 = 3^6$ , not $9^6$ . You are counting factors of 3, not multiplying the bases together.

Worked example: A power of a product

Simplify $(4m^5)^3$ .

The exponent 3 applies to both the 4 and the $m^5$ :

\begin{aligned} (4m^5)^3 &= 4^3 \cdot (m^5)^3 \\ &= 64 \cdot m^{15} \\ &= 64m^{15} \end{aligned}

A common slip is to write $4m^{15}$ and forget to cube the 4.

Worked example: Combining several properties

Simplify $\dfrac{(x^3)^4 \cdot x^5}{x^6}$ .

Work one step at a time, starting with the power of a power.

\begin{aligned} \frac{(x^3)^4 \cdot x^5}{x^6} &= \frac{x^{12} \cdot x^5}{x^6} && \text{multiply exponents: } 3 \cdot 4 = 12 \\ &= \frac{x^{17}}{x^6} && \text{add exponents: } 12 + 5 = 17 \\ &= x^{11} && \text{subtract exponents: } 17 - 6 = 11 \end{aligned}

Tip

The properties also make arithmetic easier. To find $\dfrac{3^{10}}{3^8}$ , don't compute $3^{10} = 59{,}049$ . Just subtract: $3^{10-8} = 3^2 = 9$ .