Math Core

An exponent is shorthand for repeated multiplication, and that shorthand comes with a handful of rules. Once you know them, an expression like $\dfrac{(x^3)^4 \cdot x^2}{x^5}$ takes one line to simplify instead of a page of multiplying.

What an exponent means

In the expression $a^n$ , the number $a$ is the base and $n$ is the exponent (or power). When $n$ is a positive integer, the exponent tells you how many copies of the base to multiply:

a^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ factors}}

So $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$ and $x^3 = x \cdot x \cdot x$ . A base with no exponent written has exponent $1$ : $x = x^1$ .

Every rule in this lesson comes from writing out the factors and counting them. If you ever forget a rule, expand a small example and count.

Multiplying powers with the same base

Multiply $x^3$ by $x^4$ by writing out every factor:

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

There are $3 + 4 = 7$ factors of $x$ in all. The exponents add.

Dividing powers with the same base

Divide $x^6$ by $x^2$ by writing it as a fraction and canceling pairs of factors (assume $x \ne 0$ ):

\frac{x^6}{x^2} = \frac{x \cdot x \cdot x \cdot x \cdot \cancel{x} \cdot \cancel{x}}{\cancel{x} \cdot \cancel{x}} = x^4.

Two factors cancel, leaving $6 - 2 = 4$ . The exponents subtract. (In this lesson the top exponent is always the larger one. The next lesson shows what happens when it isn't.)

Raising a power to a power

In $(x^2)^3$ , the whole block $x^2$ is used as a factor three times:

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

Adding $2$ three times is the same as $2 \cdot 3$ . The exponents multiply.

Powers of products and quotients

An exponent outside parentheses applies to every factor inside:

(3x)^2 = (3x)(3x) = 3 \cdot 3 \cdot x \cdot x = 3^2 x^2 = 9x^2.

The same works for a fraction: $\left(\dfrac{x}{5}\right)^3 = \dfrac{x}{5} \cdot \dfrac{x}{5} \cdot \dfrac{x}{5} = \dfrac{x^3}{5^3} = \dfrac{x^3}{125}$ .

Properties of exponents

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

property	rule	example
product of powers	$a^m \cdot a^n = a^{m+n}$	$5^2 \cdot 5^4 = 5^6$
quotient of powers	$\dfrac{a^m}{a^n} = a^{m-n}$	$\dfrac{y^9}{y^3} = y^6$
power of a power	$(a^m)^n = a^{mn}$	$(k^4)^2 = k^8$
power of a product	$(ab)^n = a^n b^n$	$(2p)^3 = 8p^3$
power of a quotient	$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$	$\left(\dfrac{3}{w}\right)^2 = \dfrac{9}{w^2}$

The first two rules need the same base.

Common mistake

Two mix-ups cause most mistakes:

Multiplying the bases. $2^3 \cdot 2^4 = 2^7$ , not $4^7$ . The base stays the same; only the exponents combine.
Mixing up adding and multiplying exponents. $x^2 \cdot x^5 = x^7$ (add), but $(x^2)^5 = x^{10}$ (multiply). Ask yourself: am I multiplying two powers, or raising one power to another?

Also, an exponent does not distribute over addition: $(x + 3)^2$ is not $x^2 + 9$ . The power-of-a-product rule works only for factors that are multiplied.

Putting the rules together

With coefficients, handle the numbers and each variable separately. Multiply or divide the coefficients as ordinary numbers, then apply the exponent rules to each variable.

Worked example: Product and quotient

Simplify $(4x^5)(-3x^2)$ and $\dfrac{18a^7 b^4}{6a^2 b}$ .

For the product, multiply the coefficients and add the exponents of $x$ :

(4x^5)(-3x^2) = (4 \cdot (-3))\, x^{5+2} = -12x^7.

For the quotient, divide the coefficients and subtract exponents for each variable. Remember $b = b^1$ :

\frac{18a^7 b^4}{6a^2 b} = \frac{18}{6} \cdot a^{7-2} \cdot b^{4-1} = 3a^5 b^3.

Worked example: Power of a product

Simplify $(-2m^3 n)^4$ .

Raise every factor to the fourth power, including the $-2$ :

\begin{aligned} (-2m^3 n)^4 &= (-2)^4 \cdot (m^3)^4 \cdot n^4 && \text{power of a product} \\ &= 16 \cdot m^{12} \cdot n^4 && \text{power of a power} \\ &= 16m^{12} n^4 \end{aligned}

The result is positive because an even number of negative factors multiplies to a positive.

Worked example: Several rules at once

Simplify $\dfrac{(x^3)^4 \cdot x^2}{x^5}$ .

Work from the inside out: first the power of a power, then the product, then the quotient.

\frac{(x^3)^4 \cdot x^2}{x^5} = \frac{x^{12} \cdot x^2}{x^5} = \frac{x^{14}}{x^5} = x^9.

Tip

Check a simplification by plugging in a small number. With $x = 2$ : $\dfrac{(2^3)^4 \cdot 2^2}{2^5} = \dfrac{4096 \cdot 4}{32} = 512$ , and $2^9 = 512$ . They match.

Rewriting with a common base

Sometimes the bases look different but aren't really. Since $4 = 2^2$ and $8 = 2^3$ , you can rewrite powers of $4$ and $8$ as powers of $2$ and then combine them.

Worked example: Matching the bases

Write $8^2 \cdot 4^3$ as a single power of $2$ .

8^2 \cdot 4^3 = (2^3)^2 \cdot (2^2)^3 = 2^6 \cdot 2^6 = 2^{12}.

Check: $64 \cdot 64 = 4096 = 2^{12}$ .

Practice

Practice 1

Write $x^4 \cdot x^9$ as a single power $x^n$ . What is $n$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Write $\dfrac{m^{12}}{m^4}$ as a single power $m^n$ . What is $n$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Write $(y^3)^5$ as a single power $y^n$ . What is $n$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Evaluate $\dfrac{2^3 \cdot 2^4}{2^5}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Simplify $(2x^3)^4$ .

Practice 6

Simplify $\dfrac{20p^6 q^5}{4p^2 q}$ .

Practice 7

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

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Write $2^5 \cdot 4^3$ as a single power $2^n$ . What is $n$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.