Exponents: Basic Rules - Adding, Subtracting, Dividing & Multiplying

Performing calculations and dealing with exponents forms a crucial part of higher-level math. Although expressions involving multiple exponents, negative exponents and more can seem very confusing, all of the things you have to do to work with them can be summed up by a few simple rules. Learn how to add, subtract, multiply and divide numbers with exponents and how to simplify any expressions involving them, and you'll feel much more comfortable tackling problems with exponents.

An exponent refers to the number that something is raised to the power of. For example, ​x​⁴ has 4 as an exponent, and ​x​ is the "base." Exponents are also called "powers" of numbers and really represent the amount of time a number has been multiplied by itself. So

\(x^4 = x × x × x × x\)

​__​Exponents can also be variables; for example, 4​_^x​ represents four multiplied by itself ​x_​ times.

Completing calculations with exponents requires an understanding of the basic rules that govern their use. There are four main things you need to think about: adding, subtracting, multiplying and dividing.

Adding exponents and subtracting exponents really doesn't involve a rule. If a number is raised to a power, add it to another number raised to a power (with either a different base or different exponent) by calculating the result of the exponent term and then directly adding this to the other. When you're subtracting exponents, the same conclusion applies: simply calculate the result if you can and then perform the subtraction as usual. If both the exponents and the bases match, you can add and subtract them like any other matching symbols in algebra. For example:

\(x^y + x^y = 2x^y \text{ and } 3x^y – 2x^y = x^y\)

Multiplying exponents depends on a simple rule: just add the exponents together to complete the multiplication. If the exponents are above the same base, use the rule as follows:

\(x^m × x^n = x^{m + n}\)

So if you have the problem ​x​³ × ​x​², work out the answer like this:

\(x^3 × x^2 = x^{3 + 2} = x^5\)

Or with a number in place of ​x​:

\(2^3 × 2^2 = 2^5 = 32\)

Dividing exponents has a very similar rule, except you subtract the exponent on the number you're dividing by from the other exponent, as described by the formula:

\(x^m ÷ x^n = x^{m – n}\)

So for the example problem ​x​⁴ ÷ ​x​², find the solution as follows:

\(x^4 ÷ x^2 = x^{4-2} = x^2\)

And with a number in place of the ​x​:

\(5^4 ÷ 5^2 = 5^2 = 25\)

When you have an exponent raised to another exponent, multiply the two exponents together to find the result, according to:

\((x^y)^z = x^{y×z}\)

Finally, any exponent raised to the power of 0 has a result of 1. So:

\(x^0 = 1 \text{ for any number }x\)

Use the basic rules for exponents to simplify any complicated expressions involving exponents raised to the same base. If there are different bases in the expression, you can use the rules above on matching pairs of bases and simplify as much as possible on that basis.

If you want to simplify the following expression:

\((x^{-2}y^4)^3 ÷ x^{-6}y^2\)

You'll require a few of the rules listed above. First, use the rule for exponents raised to powers to make it:

\(\begin{aligned}
(x^{-2}y^4)^3 ÷ x^{-6}y^2 &= x^{−2×3}y^{4×3}÷ x^{−6}y^2 \
&= x^{−6}y^{12} ÷ x^{−6}y^2
\end{aligned}\)

And now the rule for dividing exponents can be used to solve the rest:

\(\begin{aligned}
x^{−6}y^{12} ÷ x^{−6}y^2 &= = x^{−6-(−6)} y^{12-2} \
&= x^{−6+6} y^{12-2} \
&= x^0 y^{10} \
&= y^{10}
\end{aligned}\)