Exponent Facts for Kids

An exponent is a special number that tells you how many times to multiply a number by itself. Think of it as a shortcut for repeated multiplication!

For example, in the number $5^4$ , the number 5 is called the base, and the small number 4 is the exponent. You can read this as "5 to the power of 4". It means you multiply 5 by itself 4 times: $5 \cdot 5 \cdot 5 \cdot 5 = 3125$ .

In general, if you see $x^y$ , it means you multiply the base number $x$ by itself $y$ times.

When a number is raised to the power of two, like $x^2$ , we often say it's "squared". This is because if you have a square with sides of length $x$ , its area is $x \cdot x$ , or $x^2$ .

Similarly, when a number is raised to the power of three, like $x^3$ , we say it's "cubed". This is because a cube with sides of length $x$ has a volume of $x \cdot x \cdot x$ , or $x^3$ .

Exponents are super useful in algebra and other parts of mathematics. They help us write very long multiplication problems in a much shorter way.

Basic Rules of Exponents
Negative Exponents
Related pages
See also

Basic Rules of Exponents

There are a few simple rules that help us work with exponents. Let's look at them!

Product Rule: Multiplying Exponents

When you multiply numbers with the same base, you can add their exponents.

The rule is: $a^m \cdot a^n = a^{m+n}$

What it means:* If you have $a$ multiplied by itself $m$ times, and you multiply that by $a$ multiplied by itself $n$ times, you end up with $a$ multiplied by itself a total of $m+n$ times.

Example:*

$3^2 \cdot 3^4$ This means $(3 \cdot 3) \cdot (3 \cdot 3 \cdot 3 \cdot 3)$ . If you count all the 3s, there are 6 of them. So, $3^2 \cdot 3^4 = 3^6$ . Notice that $2 + 4 = 6$ , which matches the rule!

Quotient Rule: Dividing Exponents

When you divide numbers with the same base, you can subtract their exponents.

The rule is: $\frac{a^m}{a^n}=a^{m-n}$

What it means:* If you have $a$ multiplied $m$ times on top and $a$ multiplied $n$ times on the bottom, you can cancel out $n$ of the $a$ 's from both top and bottom. You are left with $a$ multiplied $m-n$ times.

Example:*

$\frac{4^4}{4^2}$ This means $\frac{4 \cdot 4 \cdot 4 \cdot 4}{4 \cdot 4}$ . You can cancel two 4s from the top and two 4s from the bottom. You are left with $4 \cdot 4$ , which is $4^2$ . Notice that $4 - 2 = 2$ , which matches the rule!

Sometimes, when you subtract the exponents, you might get a negative number. We'll learn about that next!

Zero Rule: Exponent of Zero

Any number (except zero) raised to the power of zero is always 1.

The rule is: $b^0 = 1$ (where $b$ is not zero)

Why it works:*

Let's use the Quotient Rule. We know that $\frac{b^m}{b^m} = b^{m-m} = b^0$ . But we also know that any number divided by itself is 1 (as long as it's not zero). So, $\frac{b^m}{b^m} = 1$ . This means $b^0$ must be equal to 1!

Negative Exponents

A negative exponent means you take the reciprocal of the number with a positive exponent. The reciprocal of a number is 1 divided by that number.

The rule is: $x^{-a}=\frac{1}{x^{a}}$

What it means:* Instead of multiplying, a negative exponent tells you to divide. For example, $x^{-1}$ means $1/x$ .

Example:*

$5^{-1} = \frac{1}{5^1} = \frac{1}{5}$ $2^{-3} = \frac{1}{2^3} = \frac{1}{2 \cdot 2 \cdot 2} = \frac{1}{8}$

Why it works:*

Let's think about the Product Rule again: $b^x \cdot b^{-x} = b^{x+(-x)} = b^0$ . We already learned that $b^0 = 1$ . So, $b^x \cdot b^{-x} = 1$ . To find out what $b^{-x}$ is, we can divide both sides by $b^x$ : $\frac{b^x \cdot b^{-x}}{b^x} = \frac{1}{b^x}$ This leaves us with $b^{-x} = \frac{1}{b^x}$ .