Exponent Rules

In this tutorial, we’ll look at the various rules that help us work with exponents and simplify expressions involving them. But before we dive into the exponent rules, here’s a quick intro to exponents.

What Are Exponents?

Exponents (also known as indices or powers) are shorthand for repeated multiplication. The exponent of a number tells us how many copies of it are multiplied together.

For example,

With this in mind, let’s move ahead.

Exponent Rules – At a Glance

Here’s a quick snapshot of all the exponent rules discussed in this tutorial. If something doesn’t make sense, don’t worry. I’ll explain each of the rules in sections that follow.

Rule	Example
$\hspace{0.2em} a^0 = 1, \hspace{0.4em} a \neq 0 \hspace{0.2em}$	$\hspace{0.2em} 2^0 = 1 \hspace{0.2em}$
$\hspace{0.2em} a^{-n} = 1/a^n \hspace{0.2em}$	$\hspace{0.2em} 2^{-3} = 1/2^3 \hspace{0.2em}$
$\hspace{0.2em} a^m \cdot a^n = a^{m + n} \hspace{0.2em}$	$\hspace{0.2em} 9^2 \cdot 9^3 = 9^5 \hspace{0.2em}$
$\hspace{0.2em} a^m/a^n = a^{m - n} \hspace{0.2em}$	$\hspace{0.2em} 9^5/9^2 = 9^3 \hspace{0.2em}$
$\hspace{0.2em} (a^m)^n = a^{mn} \hspace{0.2em}$	$\hspace{0.2em} (3^5)^2 = 3^{10} \hspace{0.2em}$
$\hspace{0.2em} a^m b^m = (ab)^m \hspace{0.2em}$	$\hspace{0.2em} 2^3 \cdot 5^3 = (2 \cdot 5)^3 \hspace{0.2em}$
$\hspace{0.2em} a^m / b^m = (a/b)^m \hspace{0.2em}$	$\hspace{0.2em} 2^3 / 5^3 = (2 / 5)^3 \hspace{0.2em}$
$\hspace{0.2em} a^{1/n} = \sqrt[n]{a} \hspace{0.2em}$	$\hspace{0.2em} 9^{1/4} = \sqrt[4]{9} \hspace{0.2em}$
$\hspace{0.2em} (-1)^n = 1$ $\hspace{0.2em} \text{ if } n \text{ is even}$	$\hspace{0.2em} (-1)^4 = 1 \hspace{0.2em}$
$\hspace{0.2em} (-1)^n = -1$ $\hspace{0.2em} \text{ if } n \text{ is odd}$	$\hspace{0.2em} (-1)^3 = -1 \hspace{0.2em}$

Rules Explained

Alright, in this section, we'll take a closer look at each of the rules mentioned in the table above.

Zero Power Rule

Any non-zero entity (number or expression) raised to the zeroth power equals $\hspace{0.2em} 1 \hspace{0.2em}$ .

a^0 = 1, \hspace{0.75em} a \neq 0

For example —

$\hspace{0.2em} 3^0 = 1 \hspace{0.2em}$
$\hspace{0.2em} (x - 2)^0 = 1, \hspace{1em} x \neq 2 \hspace{0.2em}$

Note — For the second example, we have the condition $\hspace{0.2em} x \neq 2 \hspace{0.2em}$ because if $\hspace{0.2em} x \hspace{0.2em}$ is $\hspace{0.2em} 2 \hspace{0.2em}$ , the base, $\hspace{0.2em} x - 2 \hspace{0.2em}$ , would become $\hspace{0.2em} 0$ . And that would leave us with $\hspace{0.2em} 0^0$ , which isn't equal to $\hspace{0.2em} 1 \hspace{0.2em}$ .

So, what about $\hspace{0.2em} 0^0 \hspace{0.2em}$ ? There is some disagreement about it but according to the popular opinion, the value of $\hspace{0.2em} 0^0$ should be $\hspace{0.2em} 1 \hspace{0.2em}$ , following the general trend.

Negative Exponent Rule

Negative exponents imply repeated division (instead of multiplication).

That means —

\begin{align*} 2^{-3} \hspace{0.2em} &= \hspace{0.2em} 1 \div 2^3 \\[1em] &= \hspace{0.2em} \frac{1}{2^3} \end{align*}

In general,

a^{-n} = \frac{1}{a^n}

So you can turn negative exponents into positive exponents (and the other way around) by taking the reciprocal of the base.

Here are a couple of examples.

$4^{-2} = \frac{1}{4^2}$
$\frac{1}{5^{-7}} = 5^7$

Product of Powers Rule

When multiplying exponents with the same base, add the powers and keep the same base.

a^m \cdot a^n = a^{m + n}

For example —

$2^{4} \cdot 2^5 \hspace{0.25em} = \hspace{0.25em} 2^{4 + 5} \hspace{0.25em} = \hspace{0.25em} 2^9$
$9^{3} \cdot 9^{-1} \hspace{0.25em} = \hspace{0.25em} 9^{3 + (-1)} \hspace{0.25em} = \hspace{0.25em} 9^2$

Quotient of Powers Rule

When dividing exponents with the same base, subtract the powers and keep the same base.

\frac{a^m}{a^n} = a^{m - n}

For example —

$\frac{3^5}{3^3} \hspace{0.25em} = \hspace{0.25em} 3^{5 - 3} \hspace{0.25em} = \hspace{0.25em} 3^2$
$\frac{4^2}{4^{-1}} \hspace{0.25em} = \hspace{0.25em} 4^{2 - (-1)} \hspace{0.25em} = \hspace{0.25em} 4^3$

Power of a Power Rule

When an entity with a power on it is raised to another power, we multiply the powers, keeping the same base.

(a^n)^m = a^{mn}

For example —

$(2^{3})^5 \hspace{0.25em} = \hspace{0.25em} 2^{3 \times 5} \hspace{0.25em} = \hspace{0.25em} 2^{15}$
$(9^{3})^{-2/3} \hspace{0.25em} = \hspace{0.25em} 9^{3 \times -2/3} \hspace{0.25em} = \hspace{0.25em} 9^{-2}$

Power of Products Rule

When a product is raised to a power, the power gets distributed to each factor in the product.

(ab)^n = a^n \cdot b^n

A couple of examples —

$(2 \times 10)^4 \hspace{0.25em} = \hspace{0.25em} 2^4 \times 10^4$
$(9 \times 5)^{1/2} \hspace{0.25em} = \hspace{0.25em} 9^{1/2} \times 5^{1/2}$

Power of Quotients Rule

When a quotient or an expression involving division is raised to a power, the power gets distributed to both the divisor and the dividend.

\left ( \frac{a}{b} \right ) ^n = \frac{a^n}{b^n}

A couple of examples —

$\left ( \frac{4}{7} \right )^2 \hspace{0.25em} = \hspace{0.25em} \frac{4^2}{7^2}$
$\left ( \frac{1}{5} \right )^3 \hspace{0.25em} = \hspace{0.25em} \frac{1^3}{5^3}\hspace{0.25em} = \hspace{0.25em} \frac{1}{5^3}$

Fractional Exponent Rule

Fractional exponents represent roots. The denomiantor in a fractional exponent gives the index of the root and the numerator stays the usual integral exponent.

a^{\frac{1}{n}} = \sqrt[n]{a}

Here are two examples.

$\hspace{0.2em} 8^{1/3} = \sqrt[3]{8} \hspace{0.2em}$
$\hspace{0.2em} 6^{2/5} = \sqrt[5]{6^2} \hspace{0.2em}$

Powers of –1

$-1 \hspace{0.2em}$ raised to an odd power equals $\hspace{0.2em} -1$ . And $-1 \hspace{0.2em}$ raised to an even power equals $\hspace{0.2em} 1$ .

So,

$\hspace{0.2em} (-1)^n \hspace{0.2em} = \hspace{0.2em} -1, \hspace{0.5em} \text{if} \hspace{0.5em} n \hspace{0.5em} \text{is odd} \hspace{0.2em}$
$\hspace{0.2em} (-1)^n \hspace{0.2em} = \hspace{0.2em} 1, \hspace{0.5em} \text{if} \hspace{0.5em} n \hspace{0.5em} \text{is even} \hspace{0.2em}$

Even numbers are integers that don't leave a remainder when divided by $\hspace{0.2em} 2 \hspace{0.2em}$ . In other words, they are numbers divisible by $\hspace{0.2em} 2 \hspace{0.2em}$ . For example, $\hspace{0.2em} 4 \hspace{0.2em}$ , $\hspace{0.2em} -6 \hspace{0.2em}$ , $\hspace{0.2em} 10 \hspace{0.2em}$ , etc.

Integers not divisible by $\hspace{0.2em} 2 \hspace{0.2em}$ are termed as odd numbers. For example, $\hspace{0.2em} 1 \hspace{0.2em}$ , $\hspace{0.2em} 7 \hspace{0.2em}$ , $\hspace{0.2em} -11 \hspace{0.2em}$ , etc.

Alright, a couple of examples.

$\hspace{0.2em} (-1)^6 \hspace{0.25em} = \hspace{0.25em} 1 \hspace{0.2em}$
$\hspace{0.2em} (-1)^7 \hspace{0.25em} = \hspace{0.25em} -1 \hspace{0.2em}$

And with that, we come to the end of this tutorial on laws of exponents or exponent rules. Until next time.