Math Core

Multiplication is a shortcut for adding the same number again and again. Exponents are the next shortcut: a quick way to write multiplying the same number again and again. They show up in area, volume, computer memory, population growth, and almost every formula you'll meet in Algebra 1.

Base and exponent

Instead of writing $5 \cdot 5 \cdot 5 \cdot 5$ , you can write $5^4$ . The big number is the base, the factor being repeated. The small raised number is the exponent, which counts how many times the base is used as a factor.

5^4 = \underbrace{5 \cdot 5 \cdot 5 \cdot 5}_{4 \text{ factors}} = 625

The whole expression $5^4$ is called a power. You read it as "5 to the fourth power" or "5 to the fourth."

Definition

Power

A power $a^n$ is made of a base $a$ and an exponent $n$ . When $n$ is a positive whole number,

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

Two exponents have special names because of geometry:

$a^2$ is " $a$ squared." A square with side length 6 has area $6^2 = 36$ .
$a^3$ is " $a$ cubed." A cube with edge length 4 has volume $4^3 = 64$ .

An exponent of 1 means the base appears once, so $a^1 = a$ . For example, $9^1 = 9$ . When you see a number with no exponent written, you can think of its exponent as 1.

Worked example: Reading and evaluating powers

Write each product as a power, then evaluate it.

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$
$\dfrac{1}{3} \cdot \dfrac{1}{3} \cdot \dfrac{1}{3}$
$0.4 \cdot 0.4$

Solutions.

Six factors of 2: $2^6 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64$ .
Three factors of $\dfrac{1}{3}$ : $\left(\dfrac{1}{3}\right)^3 = \dfrac{1 \cdot 1 \cdot 1}{3 \cdot 3 \cdot 3} = \dfrac{1}{27}$ .
Two factors of 0.4: $0.4^2 = 0.16$ .

Powers of 10

Powers of 10 follow an easy pattern. The exponent tells you how many zeros come after the 1.

Power	Product	Value
$10^1$	$10$	$10$
$10^2$	$10 \cdot 10$	$100$
$10^3$	$10 \cdot 10 \cdot 10$	$1{,}000$
$10^6$	six factors of 10	$1{,}000{,}000$

You'll use this pattern later in this unit to write very large and very small numbers.

Negative bases

You already know that a negative times a negative is positive. So when the base is negative, count the negative factors:

(-2)^4 = (-2)(-2)(-2)(-2) = 16 \qquad (-2)^3 = (-2)(-2)(-2) = -8

An even number of negative factors gives a positive answer. An odd number gives a negative answer.

Common mistake

Parentheses matter. In $(-3)^2$ the base is $-3$ , so $(-3)^2 = (-3)(-3) = 9$ . In $-3^2$ the base is just $3$ , and the negative sign is applied afterward: $-3^2 = -(3 \cdot 3) = -9$ . Also, $3^2$ means $3 \cdot 3$ , not $3 \cdot 2$ .

Exponents in the order of operations

Exponents come right after parentheses in the order of operations: parentheses, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

Worked example: Order of operations with exponents

Evaluate $4 + 2 \cdot 3^2$ .

Do the exponent first, then multiply, then add:

\begin{aligned} 4 + 2 \cdot 3^2 &= 4 + 2 \cdot 9 \\ &= 4 + 18 \\ &= 22 \end{aligned}

If you multiplied first, you would get $4 + 6^2 = 40$ , which is wrong. The exponent belongs only to the 3.

Worked example: Evaluating with a variable

Evaluate $x^3 - 2x^2$ when $x = -3$ .

Replace every $x$ with $(-3)$ , keeping the parentheses:

\begin{aligned} (-3)^3 - 2(-3)^2 &= -27 - 2(9) \\ &= -27 - 18 \\ &= -45 \end{aligned}

Tip

When you substitute a negative number, always put it in parentheses. That way you won't accidentally compute $-3^2 = -9$ when you meant $(-3)^2 = 9$ .