Exponents & Radicals — Core Skills

Exponents: The Basics

Part 1 of 2 — One Skill, One Idea

An exponent is the small raised number. It tells you how many times to multiply the base by itself.

$2^{3} = 2 \times 2 \times 2 = 8$

Here the base is $2$ and the exponent is $3$. We say "two to the third power."

The one move: count the factors

Most SAT exponent questions are one rule. You can get every one of them by counting factors.

Multiplying. What is $2^{3} \times 2^{4}$?

Write it out:

$2^{3} \times 2^{4} = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2)$

Count the twos. There are $3$ of them, then $4$ more. That is $7$ twos in a row.

$2^{3} \times 2^{4} = 2^{7}$

So when the bases match and you multiply, you add the exponents: $3 + 4 = 7$.

The three rules that come from counting

• Multiply → add the exponents. $x^{5} \cdot x^{2} = x^{7}$
• Divide → subtract the exponents. $\frac{x^6}{x^{}}$, because two of the 's on the bottom cancel two on the top.

One fact to memorize

Anything raised to the zero power is $1$.

$5^{0} = 1 \qquad 12^{0} = 1 \qquad x^{0} = 1$

It is not $0$. It is $1$. That one comes up often enough to be worth remembering on its own.

Roots and Negative Exponents

Part 2 of 2 — Practice

The rules from Part 1

• Multiply, same base → add the exponents. $x^{3} \cdot x^{4} = x^{7}$
• Divide, same base → the exponents.