Exponents might look small, but they pack a serious punch. With just a little symbol, you can turn a long multiplication problem into a short and quick expression.
In this easy-to-follow guide, we’ll walk through exactly what exponents are, how they work, and why they’re so useful. We’ll break down the parts of an exponent, share real-life examples, go over some cool rules, and quiz your skills along the way.
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An exponent is a way to show how many times we multiply a number by itself. At Mathnasium, we also define it as the power to which a base is raised.
Exponents are like math’s secret shorthand. They take long strings of multiplication and turn them into quick, powerful expressions that are easy to read and even more fun to use.
When we want to multiply the same number multiple times, writing it out can get messy. Imagine writing this:
\( \displaystyle 2 \times 2 \times 2 \)
That’s a lot of twos! If we had to write that out every time, it would take forever!
So instead of writing \( \displaystyle 2 \times 2 \times 2 \), we write the expression using an exponent.
We write it like this: 2³
Let’s look at the expression closer:
2 is called the base. It’s the number we’re multiplying.
3 is the exponent. It tells us how many times to multiply the base by itself.
So 2³ means: \( \displaystyle 2 \times 2 \times 2 = 8 \)
Exponents are powerful tools used in math, science, technology, games, and everyday life.
You’ll often see them used in geometry, especially when measuring area. For example, when you find the area of a square, you multiply the length of one side by itself.If a square has sides that are each 5 inches long, its area is \( \displaystyle 5 \times 5 = 25 \) square inches. (161.29 cm²)
Exponents also pop up in places you might not expect, like video games.
Suppose your score doubles every time you beat a level. After one level, your score is doubled. After two levels, it’s doubled again. By the third time, instead of writing \( \displaystyle 2 \times 2 \times 2 \) to figure it out, you can just write 2³.
That shows your points have grown to eight times your original score. Using exponents here helps us describe how fast something can grow, like your score multiplying with every win.
Another great example comes from science.
Bacteria often grow by doubling, and they can multiply really fast. If one bacterium splits into two, and then each of those two splits again, we end up with four bacteria. That’s \( \displaystyle 2 \times 2 \), or 2².
So when scientists study things that grow quickly, like germs, populations, or even money in a bank account, they often use exponents to describe that growth clearly and efficiently.
In all these situations, exponents help us describe repeated multiplication without having to write every step.
Let’s take a closer look at how to read exponents and what they really mean.
When you see something like 4², what exactly are we multiplying?
Going back to our definition, we can see that the exponent (²) tells us how many times we’re multiplying the number (4) by itself.
In this case, we have two fours in our multiplication, so 4² can also be written as 4 × 4 (and the other way around), which gives us 16.
Let’s try a few more examples.
If we write 3², the exponent (²) tells us that we have two threes in our multiplication: 3² is the same as \( \displaystyle 3 \times 3 \), so 3²=9.
Now try thinking about 2⁴. The base is 2, and the exponent is 4, so we multiply 2 by itself four times: \( \displaystyle 2 \times 2 \times 2 \times 2 \). That gives us 16.
Notice how quickly these numbers can grow when we keep multiplying the base by itself!
Imagine having to write out \( \displaystyle 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \) every time you wanted to say 2⁷. That’s a lot of work!
Exponents save us time and space by turning long strings of multiplication into something much easier to write and understand. Instead of counting every factor one by one, you just look at the exponent and know exactly how many times the base is being used.
As you get more comfortable with exponents, you’ll come across a few special rules that might seem surprising at first. But don’t worry once we go through them together, they’ll make perfect sense.
First, let’s talk about the exponent of 1.
Remember: the exponent tells us how many times to multiply the number by itself.
In the example 5², the exponent 2 means we have two fives in the multiplication: \( \displaystyle 5 \times 5 \).
So, if the exponent is 1, how many fives do we have?
Just one!
That’s why 5¹ is simply 5.
Nothing fancy here. Exponent 1 is just saying, “Use the number once.”
Whether it’s 3¹, 10¹, or 1¹, the result is always the base number.
These rules might seem like math magic at first, but they’re really just ways of keeping the patterns of multiplication consistent. When we follow these rules, exponents behave in predictable, logical ways and that helps us solve even bigger problems later on with confidence and clarity.
Now let’s look at the zero exponent.
Here’s the rule: Any number (except 0) raised to the power of 0 equals 1.
So, 7⁰ = 1.
Or 100⁰ = 1.
Even 2⁰ = 1.
This might feel strange, after all, how does multiplying a number by itself zero times give you one?
Here’s a fun way to think about it: Every time we reduce an exponent by one, we’re dividing by the base.
Let’s look at powers of 2:
23=8
22=4
21=2
20=1
Notice what happens each time we lower the exponent by 1: we divide by 2.
Going from 23 to 22, we divide 8 ÷ 2 = 4.
Going from 22 to 21, we divide 4 ÷ 2 = 2.
Going from 21 to 20, we divide 2 ÷ 2 = 1.
See the pattern?
So, when the exponent gets smaller by 1, the value gets divided by 2.
That’s why 2⁰, and any number to the power of 0, equals 1.
You’ve learned what exponents are, how they work, and even explored some cool rules. Now it’s your turn to put that knowledge into action!
Don’t worry if you get something wrong—at Mathnasium, we believe every mistake is just another step toward mastering math.
Try solving these on your own:
What is 2²?
What is 3³?
What is 4¹?
How would you write \( \displaystyle 5 \times 5 \times 5 \) using an exponent?
What is 2⁴?
What is 6⁰?
What is 7¹?
How would you write \( \displaystyle 9 \times 9 \times 9 \times 9 \) as an exponent?
What is 10²?
How would you write \( \displaystyle 4 \times 4 \)in exponent form?
What is 5³?
Once you’re done, scroll down to check your answers. No peeking!
Questions about algebraic exponents? We have your answers!
Yes! Any number can be a base, positive or negative, whole or decimal. Right now, we usually use whole numbers as exponents, but as you learn more math, you’ll explore fractions and even negative exponents too.
One big mistake is thinking the exponent means multiplication. For example, 5² does not mean \( \displaystyle 5 \times 2 \)—it means \( \displaystyle 5 \times 5 \). Always remember: you’re multiplying the base by itself, not by the exponent.
Try writing out the multiplication to double-check. For example, for 2⁴, write \( \displaystyle 2 \times 2 \times 2 \times 2 \) and see if you get 16. For bigger numbers, you can use a calculator to check your work.
Yes, and no matter what the exponent is, if the base is 1, the answer is always 1! Because 1 multiplied by itself any number of times is still 1.
That’s a tricky one! In most basic math, we leave 0⁰ undefined because it doesn’t follow the usual patterns. You’ll explore this more in advanced math later on.
When the exponent is 1, it means you only use the base once. So 5¹ just means 5. You’re not multiplying it by anything else, just using it one time.
In math, the caret symbol (^) is used to represent exponents when it’s not possible to write numbers in superscript. You might see the caret symbol in calculators, computer programs, or online math activities used like so: 2^2 which means 22, or 5^3 which means 53.
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If you’ve given our quick quiz a try, check your answers below.
4
27
4
5³
16
1
7
9⁴
100
4²
125
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