Exponents are an easy way to shorten the number of terms needed when we multiply a number or variable by itself. For example, if we wanted to multiply the number
\(3\)
five times, we could write:
\(3\times 3\times 3\times 3\times 3\)
Using exponents, however, we can write:
\({3}^{5}\)
This means we’re multiplying 3 by itself five times.
In this case, we call
\(3\)
the base, and
\(5\)
the exponent.
\({x}^{n}\)
is also called “x raised to the nth power”. Therefore, in our example, we are raising 3 to the fifth power. The exponents you’ll see in your homework and tests are most often the second and third powers, such as
\({x}^{2}\)
and
\({x}^{3}\)
.
Some exponents have special names — if you raise something to the second power, you have “squared” it. And if you raise something to the third power, you have “cubed” it.
Along with numbers, you can also use exponents on variables, as seen in
\({x}^{2}\)
and
\({x}^{3}\)
above.
But what happens when we have multiple exponents in an equation? Let’s dig deeper and discover how to handle these situations in the next section.
Exponent Rules
In a hurry? Use our algebra exponent calculator to simplify your expression and get you back on track. But it’s also worth knowing the three basic rules that govern expressions with exponents:
Product Rule:
Let’s take
\({x}^{3}{x}^{8}\)
as an example — the Product Rule says if you are multiplying two terms with the same base, you can add their exponents. The result is
\({x}^{11}\)
.
Power Rule: What happens if a second exponent appears in the expression like this:
\({({x}^{5})}^{3}\)
? The Power Rule states that you can multiply the outer and inner exponents to simplify the expression, which in this case is
\({x}^{15}\)
. This also works if you have more than one variable, meaning that
\({({x}^{2}{y}^{3})}^{5}\)
becomes
\({x}^{10}{y}^{15}\)
.
Quotient Rule: Consider this example:
\(\frac{{x}^{14}}{{x}^{6}}\)
. Can you simplify it? The Quotient Rule says that when you are dividing terms with the same base, you can subtract the exponents. In our example, this gives
\({x}^{8}\)
.
Keep in mind that anything taken to the power of zero equals 1. This means no matter how complicated an equation may seem, if it has an exponent of 0, the answer is 1.
The Cymath Advantage
Start with our exponent practice problems at the top of this page to get a handle on the rules, then click on “Solution” to see the steps. You can also download the Cymath help app and get full access to our exponent calculator with steps.
At Cymath, we believe the combination of regular practice and access to solution steps can help students discover the underlying rules of key math concepts and improve overall performance.
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