How to Teach Powers of 10 and Exponents in Fifth Grade
Your 5th grader already knows that 10 x 10 = 100 and 10 x 10 x 10 = 1,000. What they probably do not know is that mathematicians got tired of writing all those tens and invented a shorthand: 10^2 and 10^3. That shorthand — exponent notation — is one of the most important new ideas in 5th-grade math, and it unlocks everything from understanding large numbers to the basics of scientific notation. But the notation itself can be confusing if it is introduced too fast, before the underlying pattern is solid.
What the research says
Exponents are a well-documented source of misconceptions. The most common error, found across multiple studies (Pitta-Pantazi, Christou & Zachariades, 2007), is treating the exponent as a multiplier: students interpret 10^3 as 10 x 3 = 30 rather than 10 x 10 x 10 = 1,000. This error persists into middle school and even high school if it is not addressed explicitly when exponents are first introduced.
The research-supported approach is to ground exponents in repeated multiplication before introducing the notation. Students who first build the pattern through concrete examples — physically writing out 10 x 10 x 10 and counting the zeros — develop a robust understanding that resists the "multiply by the exponent" misconception.
The Common Core standards (5.NBT.A.2) expect 5th graders to explain patterns in the number of zeros when multiplying by powers of 10 and to use whole-number exponents to denote powers of 10. This is a focused standard — your child needs to master powers of 10 specifically, not exponents in general (that comes in 6th grade).
Start with the pattern, not the notation
Day 1: The repeated multiplication pattern
Write this sequence and have your child calculate each one:
| Expression | Value |
|---|---|
| 10 | 10 |
| 10 x 10 | 100 |
| 10 x 10 x 10 | 1,000 |
| 10 x 10 x 10 x 10 | 10,000 |
| 10 x 10 x 10 x 10 x 10 | 100,000 |
Now ask:
Parent: What pattern do you see?
Child: Every time you multiply by another 10, you add a zero.
Parent: How many tens are multiplied together to get 1,000?
Child: Three.
Parent: And how many zeros does 1,000 have?
Child: Three.
Parent: Is that a coincidence?
Let your child sit with this. The connection between the number of tens being multiplied and the number of zeros in the result is the foundational insight. Everything else builds on it.
Day 2: Introduce the notation
Once the pattern is solid, introduce exponent notation as a shorthand:
Parent: Writing 10 x 10 x 10 x 10 x 10 gets tedious. Mathematicians invented a shorter way. Instead of writing five tens multiplied together, we write 10^5. The small number — the 5 — tells us how many tens to multiply.
Write the table again with the notation:
| Repeated multiplication | Exponent form | Value |
|---|---|---|
| 10 | 10^1 | 10 |
| 10 x 10 | 10^2 | 100 |
| 10 x 10 x 10 | 10^3 | 1,000 |
| 10 x 10 x 10 x 10 | 10^4 | 10,000 |
| 10 x 10 x 10 x 10 x 10 | 10^5 | 100,000 |
Introduce the vocabulary: in 10^3, the 10 is the base and the 3 is the exponent. We read it as "ten to the third power" or "ten to the third."
Day 2 critical check
Immediately after introducing the notation, ask:
Parent: What is 10^4?
If your child says 40, stop. They have the multiplication misconception. Go back to the table: "10^4 means 10 x 10 x 10 x 10. How many tens? Four. What is 10 x 10 x 10 x 10?" Repeat this check over several days until the response is automatic.
Connecting to place value
Fifth graders have been working with place value since kindergarten. Exponents give them a new way to describe what they already know.
Write the place value chart with powers of 10:
| Place | Value | As a power of 10 |
|---|---|---|
| Ones | 1 | 10^0 (we will explain this) |
| Tens | 10 | 10^1 |
| Hundreds | 100 | 10^2 |
| Thousands | 1,000 | 10^3 |
| Ten thousands | 10,000 | 10^4 |
| Hundred thousands | 100,000 | 10^5 |
| Millions | 1,000,000 | 10^6 |
Parent: The ones place is 10^0. That seems weird — what does it mean to multiply zero tens together?
Child: I am not sure.
Parent: Think of it this way. Every time you go one place to the right, you divide by 10. A thousand divided by 10 is a hundred. A hundred divided by 10 is ten. Ten divided by 10 is one. So the pattern says 10^0 must equal 1. We will work more with this idea in middle school, but for now, just know that any number to the zero power equals 1.
You do not need to belabor 10^0. Mention it, explain the pattern-based reasoning, and move on.
Multiplying and dividing by powers of 10
This is the practical application that the standard emphasizes. Your child should be able to quickly compute products and quotients involving powers of 10.
Multiplying: shift the decimal right
3.45 x 10^2 = 3.45 x 100 = 345
The exponent tells you how many places to move the decimal point to the right.
Parent: What is 7.2 x 10^3?
Child: 10^3 is 1,000. Move the decimal three places right. 7,200.
Practice with a mix:
| Problem | Answer |
|---|---|
| 4.5 x 10^1 | 45 |
| 4.5 x 10^2 | 450 |
| 4.5 x 10^3 | 4,500 |
| 0.038 x 10^2 | 3.8 |
| 0.038 x 10^4 | 380 |
Dividing: shift the decimal left
6,200 / 10^2 = 6,200 / 100 = 62
The exponent tells you how many places to move the decimal point to the left.
Parent: What is 45,000 / 10^3?
Child: 10^3 is 1,000. Move the decimal three places left. 45.
The estimation connection
When your child multiplies decimals (covered in the decimal operations unit), they can use powers of 10 as a checking tool:
3.7 x 20 = 3.7 x 2 x 10^1 = 7.4 x 10 = 74
Breaking problems into a simple multiplication and a power of 10 is a strategy that will serve them well through middle school.
A gentle introduction to scientific notation
Scientific notation is formally a 6th-8th grade topic, but 5th grade is the perfect time to plant the seed. Once your child is comfortable with powers of 10, show them how scientists write very large numbers:
Parent: The distance from Earth to the Sun is about 93,000,000 miles. That is a lot of zeros. Scientists write it as 9.3 x 10^7. Can you see why?
Child: 10^7 is 10,000,000. Times 9.3 is 93,000,000. It is the same number, just written differently.
Parent: Exactly. The power of 10 handles the zeros so you can focus on the important digits.
Do not drill scientific notation at this stage. Just show a few examples so the concept is familiar when it comes back in middle school.
Common mistakes and how to address them
| Mistake | Example | Fix |
|---|---|---|
| Multiplying base by exponent | 10^3 = 30 | Go back to repeated multiplication every time. "10^3 means 10 x 10 x 10, not 10 x 3." |
| Confusing multiplication direction | 2.5 x 10^3 = 0.0025 | Ask: "Are we multiplying by a big number or dividing? Should the answer get bigger or smaller?" |
| Counting zeros incorrectly | 10^4 = 1,000 (missing a zero) | Have your child write out the multiplication: 10 x 10 x 10 x 10, then count the zeros in the product. |
| Thinking exponents only work with 10 | "You cannot write 2^3" | Briefly show that 2^3 = 2 x 2 x 2 = 8. The notation works with any base. |
Practice activities
The power tower. Write a vertical "tower" of powers of 10 from 10^0 to 10^10 on a large sheet of paper. Have your child fill in the standard form next to each one. Then quiz them randomly: point to 10^6 and ask for the value, or say "one million" and ask for the exponent.
Real-world big numbers. Find large numbers in the news, in science books, or in atlases (population of countries, distances in space, sizes of cells). Have your child express each number as a product of a single digit and a power of 10. "The population of the United States is about 330,000,000. That is about 3.3 x 10^8."
The shrinking game. Start with a large number like 5,000,000. Divide by 10 repeatedly, writing each step. 500,000... 50,000... 5,000... 500... 50... 5... 0.5... 0.05. This connects powers of 10 to decimal place value in both directions.
Red flags: when your child needs more support
- Persistently says 10^3 = 30. This misconception must be fully corrected before moving on. Return to writing out 10 x 10 x 10 by hand for every problem until it breaks.
- Cannot multiply by 10, 100, or 1,000 quickly. Powers of 10 work depends on fluent mental multiplication by these values. If your child still counts zeros on paper for 10 x 100, practice that foundational skill.
- Confuses the base and the exponent. If your child reads 10^3 as "three to the tenth," they need more exposure to the notation. Label the parts repeatedly: "The big number on the bottom is the base. The small number up top is the exponent."
When to move on
Your child is ready for 6th-grade exponent work when they can:
- Evaluate any power of 10 from 10^0 through 10^9 without writing out the multiplication
- Multiply and divide decimals by powers of 10 by shifting the decimal point
- Explain in their own words what the exponent tells you (how many times to multiply the base)
- Express large numbers as a product of a single-digit number and a power of 10
What comes next
In 6th grade, exponents expand beyond powers of 10. Your child will work with expressions like 2^4, 3^5, and eventually variable expressions like x^2. They will learn the formal rules for multiplying and dividing powers (adding and subtracting exponents), and they will encounter scientific notation as a required skill for both math and science. The strong foundation in powers of 10 built this year — especially the understanding that an exponent means repeated multiplication — is what prevents the misconceptions that plague students who were taught exponents as a set of rules to memorize.