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Discover the shortcut that turns repeated multiplication into a tiny, powerful number written up high.
Have you ever written the same number multiplied over and over, like 2 × 2 × 2 × 2 × 2? That gets pretty tiring, right? For thousands of years, mathematicians felt the same way. They searched for a shorter way to write repeated multiplication, and the result is what we now call exponents (a small number written above and to the right of a base number that tells you how many times to multiply).
Here's a quick look at how exponents appeared throughout history.
Today, exponents are everywhere — from science and engineering to computer programming and video-game graphics. Learning to write and evaluate them is one of the most useful skills you can build in math.
Before we start solving problems, let's nail down the vocabulary. Exponent notation has just a few parts, but each one matters.
Two special exponents show up a lot. When the exponent is 2, we say the number is "squared" (like the area of a square). When the exponent is 1, the value is just the base itself — nothing special happens, because you're only using the base one time.
One of the coolest things about exponents is how fast the values grow. Let's look at powers of 2 to see what happens when we keep raising the exponent by one.
Look at how the bars jump! Going from 21 to 27, the value rockets from just 2 all the way up to 128. Each time the exponent goes up by one, the value doubles. That's the power of exponents — they describe things that multiply again and again, growing much faster than addition ever could.
Now let's get hands-on. Writing and evaluating exponents means two things: turning repeated multiplication into exponent form, and turning exponent form back into a single number.
When you see 43, you can say "four to the third power," "four cubed," or "four raised to the three." When you see 72, you say "seven squared" or "seven to the second power."
The zero-exponent rule might seem strange at first. Here's one way to think about it: look at the pattern 23 = 8, 22 = 4, 21 = 2. Each time the exponent drops by one, you divide by 2. So 20 = 2 ÷ 2 = 1. The pattern keeps working!
To evaluate (find the value of) an expression with exponents, follow these steps:
1. Identify the base and exponent. 2. Write out the expanded form (all the multiplications). 3. Multiply from left to right to get the standard form.
Some powers show up so often in math and science that it's helpful to memorize them. Here's a handy reference table.
| Expression | Expanded Form | Value | Read As |
|---|---|---|---|
| 22 | 2 × 2 | 4 | two squared |
| 23 | 2 × 2 × 2 | 8 | two cubed |
| 24 | 2 × 2 × 2 × 2 | 16 | two to the fourth |
| 25 | 2 × 2 × 2 × 2 × 2 | 32 | two to the fifth |
| 32 | 3 × 3 | 9 | three squared |
| 33 | 3 × 3 × 3 | 27 | three cubed |
| 42 | 4 × 4 | 16 | four squared |
| 52 | 5 × 5 | 25 | five squared |
| 102 | 10 × 10 | 100 | ten squared |
| 103 | 10 × 10 × 10 | 1,000 | ten cubed |
| 106 | 10 × 10 × 10 × 10 × 10 × 10 | 1,000,000 | ten to the sixth |
Notice something about powers of 10? The exponent tells you exactly how many zeros to write after the 1. That's why scientists love using powers of 10 to describe really big or really small quantities.
The diagram above shows every part of an exponent expression. The base (3) sits on the main line. The exponent (4) floats up high to the right. Together, they tell you to write 3 as a factor four times: 3 × 3 × 3 × 3. When you multiply that out, you get the standard form: 81.
Let's walk through a complete problem together so you can see every step.
Exponents are simple in concept but easy to mix up if you aren't careful. Let's compare what students often do wrong with what's actually correct.
| Common Mistake | Why It's Wrong | Correct Approach |
|---|---|---|
| 3⁴ = 12 | This multiplies the base by the exponent (3 × 4). Exponents aren't multiplication — they're repeated multiplication. | 3⁴ = 3 × 3 × 3 × 3 = 81 |
| 3⁴ = 3 + 3 + 3 + 3 = 12 | This adds the base four times. That's just 3 × 4 again. Exponents use multiplication, not addition. | 3⁴ = 3 × 3 × 3 × 3 = 81 |
| 2 + 5² = 7² = 49 | You must evaluate the exponent before adding. The exponent applies only to the 5, not to 2 + 5. | 2 + 5² = 2 + 25 = 27 |
| 5⁰ = 0 | Any non-zero number to the zero power is 1, not 0. The exponent tells you "no factors," and the multiplicative identity is 1. | 5⁰ = 1 |
You've learned how to write and evaluate exponents using whole numbers. But this is just the beginning! Here's a peek at where exponents go in later grades.
| What You Know Now | What's Coming Later |
|---|---|
| Whole-number exponents (like 25) | Negative exponents (like 2−3 = ¹⁄₈) and fractional exponents (like 9½ = 3) |
| Evaluating expressions with numbers | Using exponents with variables (like x3) in algebra |
| Powers of 10 (102, 103…) | Scientific notation — writing very big or very tiny numbers like 3.2 × 108 |
| Order of operations with exponents | Exponent rules for multiplying and dividing powers, like am × an = am+n |
Everything you're learning now — identifying bases, evaluating step by step, and following the order of operations — is the foundation for all of those topics. The better you understand exponents today, the easier those future lessons will be.
Time to try some on your own! Start from the top and work your way down. Each problem is a little harder than the last. Click "Show Answer" when you're ready to check.
An exponent is a shorthand way to write repeated multiplication. Every exponent expression has two parts: the base (the number being multiplied) and the exponent (how many times to use the base as a factor). For example, 5³ means 5 × 5 × 5, which equals 125. Two special cases are worth remembering: any number to the first power equals itself, and any non-zero number to the zero power equals 1.
When exponents appear alongside other operations, always evaluate the exponents first (following the order of operations). The biggest mistake to avoid is treating the exponent as regular multiplication (like 3⁴ = 12 instead of 81). If you're ever unsure, write out the expanded form — seeing all the multiplication signs keeps you on track. With practice, reading and evaluating exponent expressions will become as natural as basic multiplication itself!