Negative Exponents Explained

Math Skills: What are negative exponents? The Negative Exponent Rule, How to Do Negative Exponents

What Does a Negative Exponent Mean?

Before we talk about negative exponents, let’s quickly review what we already know about positive exponents.

For example, we know that:

Note that positive exponents take the base number and make them larger.

But, what would happen if an exponent is negative? Let’s find out by changing the exponent (2) from the previous example and making it negative (-2) as follows:

Your initial response to seeing a negative exponent is to assume that you have to evaluate the exponent and then express the result as a negative number, but that would be incorrect.

What negative exponents actually do is they take the base number and make them smaller. How does that work? When you raise a base number to a negative exponent, you are actually taking the reciprocal of the base value (expressing it as a fraction where the numerator is 1 and the denominator is the base value) and raising it to the positive form of whatever the negative exponent is.

Now let’s compare 5² and 5⁻² using the table below. What do you notice?

Now we can answer the question, what does a negative exponent mean?

In math, a negative exponent tells you to rewrite the base value as a fraction with a numerator of 1 and then raise that fraction to the positive version of the exponent.

Here are few more examples of what happens when you raise a base value to a negative exponent.

Notice negative exponents end up shrinking the base value (rather than expanding them the way that positive exponents do).

Negative Exponent Rule

After looking at the examples of negative exponents in the previous section, you may have noticed a pattern. This is because negative exponents all follow a this simple, known as the negative exponent rule.

Definition: The negative exponent rule says that any nonzero number, a, raised to a negative exponent, n, equals one over that number (as a fraction) raised to the positive version of the exponent.

The negative exponent rule applies to any nonzero number raised to an exponent, including whole numbers, decimals, fractions, and even variables like x and y.

The table below shows how the negative exponent rule applies to 2⁻³ (compared to 2³).

How to Do Negative Exponents: Examples

Now you are ready to learn how to do negative exponents using the negative exponent rule to solve a few practice problems.

Strategy for How to Do Negative Exponents:

• Step 1: Start with an expression such as 4⁻²

• Step 2: Apply the negative exponent rule and rewrite the expression as a fraction with a numerator of one and a positive exponent:

• 4⁻² = 1 / 4²

• Step 3: Evaluate: 1 / 4² = 1/16

So, 4⁻² = 1/16 or 0.0625

How to Solve Negative Exponents Example #1

Now, let’s learn how to solve negative exponents by evaluating a negative exponent other than -2.

Problem: Simplify 2⁻³

We can start off by applying the negative exponent rule to 2⁻³ by rewriting it as a fraction with a numerator of one and changing the sign of the exponent from negative to positive:

• 2⁻³ = 1/2³

Next, we can evaluate as follows:

• 1/2³ = 1/8 = 0.125

Final Answer: 2⁻³ = 1/8 or 0.125

Note: Whenever you are dealing with a negative exponent, your final answer will always be a fraction or a decimal).

How to Solve Negative Exponents Example #2

Next, let’s apply the negative exponent rule to a variable (x) being raised to a negative exponent.

Problem: Simplify x⁻⁵

Just like the previous example, start by applying the negative exponent rule and rewriting x⁻⁵ as a fraction with a numerator of one and a positive exponent:

• x⁻⁵ = 1/x⁵

Since our base value in this example is a variable (x), we can not evaluate any further.

Final Answer: x⁻⁵ = 1/x⁵

Note: While we didn’t do much in terms of “solving” this problem, rewriting variables with negative exponents in this format is often helpful when you have to simplify an algebraic expression.

How to Solve Negative Exponents Example #3

Finally, let’s learn how to apply the negative exponent rule when the base value is a fraction.

Problem: Simplify (3/4)⁻²

Whenever you have a fraction raised to a negative exponent, the first step is to flip the fraction (swap numerator and the denominator) and then raise the fraction to the positive version of the exponent”

• (3/4)⁻² = (4/3)² = 4²/3² = 16/9

Final Answer: (3/4)⁻² = 16/9 or approximately 1.78

Note: Raising a proper fraction to a negative exponent does not shrink the base value (in fact, it expands it).

Why Are Negative Exponents Important?

Now that you understand what negative exponents mean and how to use the negative exponent rule, let’s conclude by sharing some reasons why negative exponents are important.

Since negative exponents tend to shrink the base value, they are extremely useful for expressing super small numbers, which is why they are so common in scientific notation. Negative exponents are also common at higher-levels of algebra and calculus, because they allow you to rewrite and simplify complex expressions.

Outside of the classroom, negative exponents have many real-life applications including computer science, scientific measurements, and exponential decay.

By learning how to do negative exponents and how to use the negative exponent rule, you are better able to be successful in algebra and beyond.

Negative Exponents Worksheet

Are you looking for some extra practice with negative exponents and using the negative exponent rule? Click the link below to download your free Negative Exponents PDF Worksheet, which includes twenty practice problems and a complete answer key.

We recommend that you work through all of the problems on your own and them compare them to the key after you have finished. If you answer 16 or more of the questions correctly, then you are in good shape. However, if you get fewer than 16 problems correct, we highly recommend that you go back and work through this guide again.

Click Here to Download Your Negative Exponents Practice Worksheet

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