A power of 10 = 10 multiplied by itself.
Exponent form: 104 (base = 10, exponent = 4)
Expanded form: 10 × 10 × 10 × 10
Standard form (value): 10,000
Pattern: The exponent = the number of zeros in the value.
Powers of 10
Multiply & Divide with the Power of Patterns
📚
Subject
Math
⏱️
Duration
45 min
🎯
Standard
5.NBT.A.2
📋 Standards & Objectives
📜Standards
5.NBT.A.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.
5.NBT.A.1Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
🎯SWBAT
- Identify the base, exponent, expanded form, and value of a power of 10
- Multiply whole numbers and decimals by powers of 10 by moving the decimal point to the right
- Divide whole numbers and decimals by powers of 10 by moving the decimal point to the left
- Explain the pattern in zeros and decimal placement when multiplying or dividing by powers of 10
📖 Key Vocabulary
📝Exponent
The small number written above and to the right of the base. It tells you how many times to multiply the base by itself.
In 104, the exponent is 4 — it means multiply 10 four times: 10 × 10 × 10 × 10.
In 102, the exponent is 2 — it tells us 10 × 10 = 100.
📝Base
The number being multiplied repeatedly. In a power of 10, the base is always 10.
In 103, the base is 10 — the number we keep multiplying.
The base sits on the bottom, just like the base of a building holds everything up!
📝Power of 10
A number you get by multiplying 10 by itself a certain number of times. Written in exponent form like 10n.
101 = 10, 102 = 100, 103 = 1,000 — these are all powers of 10.
The number 1,000,000 is a power of 10 because 106 = 1,000,000.
📝Expanded Form
Writing a power of 10 as repeated multiplication. Shows every factor of 10 being multiplied.
The expanded form of 104 is 10 × 10 × 10 × 10.
103 in expanded form is 10 × 10 × 10 — you can see all three factors!
📝Standard Form
The regular way we write a number using digits. The final value after you compute the multiplication.
103 = 1,000 — the number 1,000 is the standard form.
When we write 100,000 instead of 105, we are writing in standard form.
🚀 How BIG Can You Go?
Think about this: if you started with the number 1 and multiplied by 10 over and over, how quickly would the numbers get enormous?
The Power of 10
1 → 10 → 100 → 1,000 → 10,000 → 100,000 → 1,000,000
Each time we multiply by 10, we just add a zero. Today we'll learn the shortcut that makes working with these giant (and tiny!) numbers simple.
By the end of this lesson, you'll be able to multiply AND divide any number by a power of 10 — using patterns instead of long multiplication.
👨🏫 What IS a Power of 10?
When we multiply 10 by itself multiple times, we can write it in a shorter way using an exponent.
10 × 10 × 10 × 10
=
104
↑
Expanded Form
Expanded Form
↑
Exponent Form
Exponent Form
The base is 10 (the number being multiplied). The exponent is 4 (how many times we multiply).
👨🏫 The Powers of 10 Table
Power of 10
Expanded Form
Value
100
Any number to the 0 power = 1
1
101
10
10
102
10 × 10
100
103
10 × 10 × 10
1,000
104
10 × 10 × 10 × 10
10,000
105
10 × 10 × 10 × 10 × 10
100,000
106
10 × 10 × 10 × 10 × 10 × 10
1,000,000
Notice the pattern: the exponent tells you how many zeros are in the standard form value!
📓 Powers of 10 Pattern
Write this in your notebook!
Key Terms
Exponent
Base
Power of 10
Notes
👨🏫 Three Ways to Write It
Every power of 10 can be written three ways. Let's practice reading them!
105
Exponent Form: 105
Read as: "ten to the fifth power"
Expanded Form:
10 × 10 × 10 × 10 × 10
Standard Form: 100,000
102
Exponent Form: 102
Read as: "ten to the second power"
Expanded Form:
10 × 10
Standard Form: 100
💡 Click examples to expand, or use J/K keys
✅ Quick Check
In 106, identify:
What is the base? What is the exponent? How many zeros in the standard form?
👍 Thumbs up when you have all three answers
Base = 10 | Exponent = 6 | Zeros = 6 (1,000,000)
💬 Turn & Talk
🤔Discuss with a Partner
Look at the Powers of 10 table we just learned. What pattern do you notice as the exponent increases by 1? What happens to the value each time?
Sentence starter: "I notice that every time the exponent goes up by 1, the value ___."
👨🏫 Whole Numbers × Powers of 10
When you multiply a whole number by a power of 10, just add zeros equal to the exponent!
7
×
103
=
7,000
The exponent is 3 → add 3 zeros after the 7
The Rule
Write the whole number, then add as many zeros as the exponent tells you.
👨🏫 Example: 35 × 10³
35
×
103
1 Write the whole number: 35
2 The exponent is 3, so add 3 zeros
3 35 + 000 = 35,000
Yes! 10³ = 1,000, and 35 × 1,000 = 35,000 ✓
📓 Multiply Whole Numbers Rule
Write this in your notebook!
Key Word
Multiply Whole Numbers × 10ⁿ
Notes
Rule: Write the whole number, then add zeros.
The exponent = the number of zeros to add.
Example: 35 × 10³
Step 1: Write 35
Step 2: Exponent is 3 → add 3 zeros
Step 3: 35,000
The exponent = the number of zeros to add.
Example: 35 × 10³
Step 1: Write 35
Step 2: Exponent is 3 → add 3 zeros
Step 3: 35,000
✅ Quick Check
What is 12 × 10⁴?
Think: write 12, then add ___ zeros.
12 + 4 zeros = 120,000
👥 Guided Practice: 8 × 10⁴
8
×
104
=
?
Work with a partner: What is the exponent? How many zeros do we add?
Exponent = 4 → add 4 zeros
8 × 10⁴ = 80,000
👥 Guided Practice: 52 × 10²
52
×
102
=
?
Your turn! Write 52, then add zeros equal to the exponent.
52 + 2 zeros = 5,200
Check: 52 × 100 = 5,200 ✓
🔄 Now With Decimals!
What We Just Learned
Whole number × power of 10 → just add zeros equal to the exponent.
But what happens when the number has a decimal point? We can't just slap zeros onto 3.45 — we need a different strategy.
What's Next
Instead of adding zeros, we move the decimal point to the right. The exponent still tells you how many places!
👨🏫 Decimals × Powers of 10
When you multiply a decimal by a power of 10, move the decimal point to the RIGHT.
Multiply → Bigger → Move RIGHT →
The exponent tells you how many places to move.
Think about it: multiplying makes a number bigger, so the decimal moves right to create a larger number. If you run out of digits, fill in zeros (this is called "annexing a zero").
👨🏫 Example: 3.45 × 10²
3.45
×
102
1 Exponent is 2 → move decimal 2 places RIGHT
2 3.45 → 345. → 345
✓ 3.45 × 10² = 345
10² = 100 → 3.45 × 100 = 345 ✓
👨🏫 Example: 7.8 × 10³
7.8
×
103
1 Exponent is 3 → move decimal 3 places RIGHT
2 7.8 → 78. → but we need 3 places total!
3 Annex zeros to fill empty spots: 7800 → wait, only need 2 more places: 7,800
✓ 7.8 × 10³ = 7,800
Annexing a zero means adding a zero as a placeholder when you run out of digits to move past.
📓 Multiply Decimals Rule
Write this in your notebook!
Key Word
Decimal × 10ⁿ
Move RIGHT
Notes
Multiply = move decimal to the RIGHT →
The exponent = how many places to move.
Example: 3.45 × 10² = 345 (moved 2 right)
Example: 7.8 × 10³ = 7,800 (moved 3 right, annexed zeros)
Remember: If you run out of digits, add zeros!
The exponent = how many places to move.
Example: 3.45 × 10² = 345 (moved 2 right)
Example: 7.8 × 10³ = 7,800 (moved 3 right, annexed zeros)
Remember: If you run out of digits, add zeros!
✅ Quick Check
What is 4.56 × 10³?
Move the decimal ___ places to the ___.
Move 3 places RIGHT → 4.56 → 45.6 → 456. → 4,560
4,560 (annexed 1 zero)
👥 Guided Practice: 12.6 × 10²
12.6
×
102
=
?
Which direction? How many places? Do we need to annex zeros?
Move decimal 2 places RIGHT
12.6 → 126. → 1,260 (annex 1 zero)
12.6 × 10² = 1,260
👥 Guided Practice: 0.45 × 10⁴
0.45
×
104
=
?
This one starts small! Move the decimal 4 places right — you'll need to annex some zeros.
0.45 → 4.5 → 45. → 450. → 4,500
Moved 4 places right (annexed 2 zeros)
0.45 × 10⁴ = 4,500
💬 Turn & Talk
🤔Discuss with a Partner
We learned two methods: adding zeros (for whole numbers) and moving the decimal right (for decimals). Are these really the same strategy? Why or why not?
Sentence starter: "I think they are the same / different because ___."
🔄 Now We Divide!
What We Just Learned
Multiply by a power of 10 → move decimal to the RIGHT → (number gets bigger)
Division is the opposite of multiplication. So if multiplying moves the decimal right...
What's Next
Divide by a power of 10 → move decimal to the ← LEFT (number gets smaller)
👨🏫 Whole Numbers ÷ Powers of 10
When you divide a number by a power of 10, move the decimal point to the LEFT ←.
400
÷
102
=
4
Move decimal 2 places LEFT: 400. → 40. → 4. = 4
Remember: Whole numbers have a "hidden" decimal at the end. 400 is really 400. — then move left!
👨🏫 Example: 5,600 ÷ 10²
5,600
÷
102
1 Place the decimal: 5,600.
2 Exponent is 2 → move decimal 2 places LEFT ←
3 5,600. → 560. → 56.
✓ 5,600 ÷ 10² = 56
👨🏫 Example: 9 ÷ 10³
9
÷
103
1 Place the decimal: 9.
2 Move 3 places LEFT ← ... but there's only 1 digit!
3 Add zeros in front: 9. → .9 → .09 → .009
✓ 9 ÷ 10³ = 0.009
Just like annexing zeros when multiplying, we add placeholder zeros to the left when dividing!
📓 Divide by Powers of 10
Write this in your notebook!
Key Word
÷ 10ⁿ
Move LEFT ←
Notes
Divide = move decimal to the LEFT ←
The exponent = how many places to move.
Example: 5,600 ÷ 10² = 56 (moved 2 left)
Example: 9 ÷ 10³ = 0.009 (moved 3 left, added zeros)
Remember: Add zeros to the LEFT if you run out of digits!
The exponent = how many places to move.
Example: 5,600 ÷ 10² = 56 (moved 2 left)
Example: 9 ÷ 10³ = 0.009 (moved 3 left, added zeros)
Remember: Add zeros to the LEFT if you run out of digits!
✅ Quick Check
What is 250 ÷ 10²?
Which direction do you move the decimal?
Divide → move LEFT 2 places
250. → 25. → 2.5
250 ÷ 10² = 2.5
👥 Guided Practice: 730 ÷ 10²
730
÷
102
=
?
Where is the hidden decimal? Which way do you move?
730. → move 2 places LEFT
730. → 73. → 7.3
730 ÷ 10² = 7.3
👨🏫 Decimals ÷ Powers of 10
Same rule applies to decimals! Divide → move decimal to the LEFT ←.
63.7
÷
101
=
6.37
1 Exponent is 1 → move decimal 1 place LEFT ←
✓ 63.7 → 6.37
The decimal just "hops" one spot to the left. The number got 10 times smaller!
👨🏫 Example: 84.5 ÷ 10²
84.5
÷
102
1 Move decimal 2 places LEFT ←
2 84.5 → 8.45 → 0.845
✓ 84.5 ÷ 10² = 0.845
10² = 100 → 84.5 ÷ 100 = 0.845 ✓
📓 The Big Rule
Write this in your notebook!
Key Words
× = RIGHT →
÷ = LEFT ←
The Big Rule
MULTIPLY by 10ⁿ → decimal moves RIGHT → (bigger)
DIVIDE by 10ⁿ → decimal moves ← LEFT (smaller)
The exponent ALWAYS = the number of places you move.
Memory trick: "Multiply Makes More" → move RIGHT to make bigger!
DIVIDE by 10ⁿ → decimal moves ← LEFT (smaller)
The exponent ALWAYS = the number of places you move.
Memory trick: "Multiply Makes More" → move RIGHT to make bigger!
👥 Guided Practice: 6.2 ÷ 10³
6.2
÷
103
=
?
Divide → which direction? How many places? Will you need to add zeros?
Move 3 places LEFT
6.2 → 0.62 → 0.062 → 0.0062
6.2 ÷ 10³ = 0.0062
💬 Turn & Talk
🤔Discuss with a Partner
Compare these two problems:
4.5 × 10² vs. 4.5 ÷ 10²
Without solving, predict: which answer is bigger? Which is smaller? How do you know?
Sentence starter: "The answer to ___ is bigger because multiplying moves the decimal ___, which makes the number ___."
👨🏫 Patterns: Powers of 10
Watch what happens when we multiply the same number by increasing powers of 10:
Expression
Value
2.5 × 100
2.5
2.5 × 101
25
2.5 × 102
250
2.5 × 103
2,500
2.5 × 104
25,000
Each time the exponent goes up by 1, the value gets 10 times bigger. The decimal point shifts one more place to the right!
👥 Pattern Practice
Complete this pattern together. Start with 8.16 and multiply by increasing powers of 10:
8.16 × 100 = 8.16
8.16 × 101 = ?
8.16 × 102 = ?
8.16 × 103 = ?
8.16 × 104 = ?
8.16 → 81.6 → 816 → 8,160 → 81,600
Each answer is 10× the one before it!
🔍 You Try: Mixed Practice
Multiply AND divide!
Multiply (→ Right)
A) 14 × 10³ = ?
B) 6.09 × 10² = ?
C) 0.7 × 10⁵ = ?
A) 14,000
B) 609
C) 70,000
Divide (← Left)
D) 3,200 ÷ 10² = ?
E) 47.5 ÷ 10¹ = ?
F) 8 ÷ 10⁴ = ?
D) 32
E) 4.75
F) 0.0008
💡 Solve all six, then reveal to check. Use J/K to focus.
🔍 Find the Missing Power of 10
What exponent makes it work?
Figure out which power of 10 is missing. Count how many places the decimal moved!
1) 3.7 × 10? = 370 (How many places did the decimal move right?)
2) 0.56 × 10? = 5,600 (Count the jumps!)
3) 92.4 × 10? = 924 (Just one little hop?)
4) 15.8 × 10? = 158,000 (That's a big jump!)
1) 102 2) 104 3) 101 4) 104
🔍 Word Problems
Problem A
A stadium holds 15 × 10³ fans. How many fans is that in standard form?
15 × 10³ = 15 + 3 zeros = 15,000 fans
Problem B
A scientist has 4,200 mL of solution. She divides it equally into 10² beakers. How many mL in each?
4,200 ÷ 10² = move 2 left
4,200 → 42.0 = 42 mL per beaker
💡 Click to focus on one problem at a time. Use J/K keys.
📓 Summary Note
Write your 1-sentence summary!
Write 1 Sentence
In the bottom of your notebook page, write one sentence explaining what you learned today about powers of 10.
Try to include: the words "multiply," "divide," "right," "left," and "exponent" in your sentence.
🎫 Exit Ticket
Show what you know!
1️⃣Multiply
23.6 × 10³ = ?
Move 3 places RIGHT
23.6 → 236 → 2,360 → 23,600
23,600
2️⃣Divide
519 ÷ 10² = ?
Move 2 places LEFT
519. → 51.9 → 5.19
5.19
3️⃣Missing Power
7.04 × 10? = 70,400
7.04 → 70,400 = moved 4 places right
10⁴
⭐ Great Work Today!
Today's Takeaways
Multiply by 10ⁿ → move decimal RIGHT → (n places)
Divide by 10ⁿ → move decimal ← LEFT (n places)
The exponent ALWAYS tells you how many places to move!
Now you're ready to tackle the practice sheet. Use the decimal movement rules and you'll crush it!