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. $\frac{x^{6}}{x^{2}} = x^{4}$, because two of the $x$'s on the bottom cancel two on the top.
• Power of a power → multiply the exponents. $(x^{3})^{2} = x^{6}$, because $x^{3}$ is written down twice, giving $3 + 3 = 6$ factors.

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 → subtract the exponents. $\frac{x^{9}}{x^{4}} = x^{5}$
• Power of a power → multiply the exponents. $(x^{2})^{5} = x^{10}$
• Zero exponent → the answer is $1$. $x^{0} = 1$

Square roots

A square root asks: what number times itself gives this? The symbol is $\sqrt{\phantom{x}}$.

$\sqrt{36} = 6 \quad \text{because} \quad 6 \times 6 = 36$

It helps to know these by sight: $\sqrt{4} = 2$, $\sqrt{9} = 3$, $\sqrt{16} = 4$, $\sqrt{25} = 5$, $\sqrt{36} = 6$, $\sqrt{49} = 7$, $\sqrt{64} = 8$, $\sqrt{81} = 9$, $\sqrt{100} = 10$.

Negative exponents

A negative exponent means one over that power. Flip it, and the exponent turns positive.

$3^{-2} = \frac{1}{3^{2}} = \frac{1}{9}$

A negative exponent does not make the answer negative. It makes it a fraction.

Fraction exponents

An exponent of $\frac{1}{2}$ is another way to write a square root.

$x^{1/2} = \sqrt{x} \qquad \text{so} \qquad 25^{1/2} = \sqrt{25} = 5$

The bottom number of the fraction is the root. So $x^{1/3} = \sqrt[3]{x}$, the cube root.