A power of 10 = 10 multiplied by itself.
Exponent form: 104 (base = 10, exponent = 4)
Expanded form: 10 × 10 × 10 × 10
Standard form: 10,000
Pattern: The exponent = the number of zeros in the value.
Powers of 10: The Complete Picture
Exponents, Decimals, Multiply & Divide — All Connected
📚
Subject
Math
⏱️
Duration
60+ min
🎯
Standard
5.NBT.A.1 / 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.
5.NBT.A.3Read, write, and compare decimals to thousandths.
🎯SWBAT
- Identify the base, exponent, expanded form, and standard form of a power of 10
- Multiply whole numbers and decimals by powers of 10 using the "move decimal RIGHT" rule
- Divide whole numbers and decimals by powers of 10 using the "move decimal LEFT" rule
- Explain WHY adding zeros and moving the decimal are the same strategy
- Use place value understanding to connect exponents, standard form, and decimal movement
📖 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.
📝Power of 10
A number you get by multiplying 10 by itself a certain number of times. Written as 10n or in standard form like 100, 1,000, etc.
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.
📝Decimal Point
The dot in a number that separates the whole number part from the fractional part. In 3.25, the dot between 3 and 2 is the decimal point.
In 4.7, the decimal point separates the 4 (ones) from the 7 (tenths).
Every whole number has a hidden decimal point at the end: 45 = 45.
📝Place Value
The value of a digit based on its position in a number. Moving left = 10× bigger. Moving right = 10× smaller.
In 555, the first 5 = 500, the second 5 = 50, and the third 5 = 5. Same digit, different place values.
In 2.34, the 3 has a place value of 3 tenths, and the 4 is 4 hundredths.
📝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're writing in standard form.
🚀 How Powerful Is 10?
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. But what happens when we multiply decimals by 10? We can't just add zeros to 3.45...
Today we'll learn one unified system that handles whole numbers AND decimals — multiply AND divide — 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
=
10,000
↑
Expanded Form
Expanded Form
↑
Exponent Form
Exponent Form
↑
Standard Form
Standard Form
The base is 10 (the number being multiplied). The exponent is 4 (how many times). The standard form is 10,000 (the value).
👨🏫 The Powers of 10 Table
Exponent
Expanded Form
Standard
Zeros
100
Any number to the 0 power
1
0
101
10
10
1
102
10 × 10
100
2
103
10 × 10 × 10
1,000
3
104
10 × 10 × 10 × 10
10,000
4
105
10 × 10 × 10 × 10 × 10
100,000
5
106
10 × 10 × 10 × 10 × 10 × 10
1,000,000
6
The exponent = the number of zeros in the standard form. This is the pattern that makes everything work!
📓 Powers of 10 Pattern
Write this in your notebook!
Key Terms
Exponent
Base
Power of 10
Notes
✅ 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. 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.
👨🏫 Two Ways — Same Answer
Exponent form and standard form are connected!
Exponent Way
35 × 103
Exponent = 3 → add 3 zeros
35,000
Standard Way
35 × 1,000
1,000 has 3 zeros → add 3 zeros
35,000
103 = 1,000. The exponent tells you the zeros, and the standard form shows them. Same math, two notations!
💡 Click examples to expand, or use J/K keys
👨🏫 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 ✓
👨🏫 The "Count the Zeros" Rule
This works whether you see exponents or standard numbers
× 10 (101)
1 zero
5 × 10 = 50
× 100 (102)
2 zeros
5 × 100 = 500
× 1,000 (103)
3 zeros
5 × 1,000 = 5,000
Zeros in the power of 10 = zeros you add to your answer
📓 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 (or 35 × 1,000)
Step 1: Write 35
Step 2: Exponent is 3 → add 3 zeros
Step 3: 35,000
Either way works: count the exponent OR count the zeros in the standard number!
The exponent = the number of zeros to add.
Example: 35 × 10³ (or 35 × 1,000)
Step 1: Write 35
Step 2: Exponent is 3 → add 3 zeros
Step 3: 35,000
Either way works: count the exponent OR count the zeros in the standard number!
✅ Quick Check: Whole Numbers
Solve these in your head!
12 × 10⁴ = ?
12 + 4 zeros = 120,000
9 × 100 = ?
100 has 2 zeros → 900
6 × 1,000 = ?
1,000 has 3 zeros → 6,000
🔄 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 — that would give us 3.45000, which is the same number!
What's Next
Instead of adding zeros, we move the decimal point to the right. The exponent (or number of zeros) still tells you how many places!
👨🏫 Decimal × 10
Watch what happens to the decimal point
1.2
×
10
=
12
10 has 1 zero → move the decimal point 1 place RIGHT
1.2 → 12.
The decimal jumped 1 spot to the right!
3.45
×
10
=
34.5
👨🏫 Decimal × 100 and × 1,000
More zeros = more places to move
1.2
×
100
=
120
← 2 zeros → move 2 right
1.2
×
1,000
=
1,200
← 3 zeros → move 3 right
0.045
×
1,000
=
45
← 3 zeros → move 3 right
When you run out of digits, fill with zeros! 1.2 × 100 → move 2 right → 120. 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 ✓ Same answer either way!
👨🏫 Example: 7.8 × 10³
What happens when you run out of digits?
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
✓ 7.8 × 10³ = 7,800
Annexing a zero means adding a zero as a placeholder when you run out of digits to move past.
⭐ Rule #1: Multiply by a Power of 10
Remember Mr. DL!
MULTIPLY = Move Decimal RIGHT →
Count the zeros (or read the exponent).
Move the decimal point that many places to the RIGHT →
× 10 (10¹)
→ 1 place
× 100 (10²)
→→ 2 places
× 1,000 (10³)
→→→ 3 places
📓 Multiply Decimals Rule
Write this in your notebook!
Key Words
Decimal × 10ⁿ
Move RIGHT
Notes
Rule #1 — Multiplying by a Power of 10:
1. Count the zeros (or read the exponent)
2. Move the decimal point that many places RIGHT →
3. Fill empty spots with zeros (annex zeros)
Examples:
3.45 × 10² = 345 (moved 2 right)
7.8 × 10³ = 7,800 (moved 3 right, annexed zeros)
Multiplying makes numbers BIGGER → decimal goes RIGHT
1. Count the zeros (or read the exponent)
2. Move the decimal point that many places RIGHT →
3. Fill empty spots with zeros (annex zeros)
Examples:
3.45 × 10² = 345 (moved 2 right)
7.8 × 10³ = 7,800 (moved 3 right, annexed zeros)
Multiplying makes numbers BIGGER → decimal goes RIGHT
✅ 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)
🔗 The Big Connection
"Adding Zeros" and "Moving the Decimal" Are the SAME Thing!
When we multiply a whole number like 35 × 1,000 and "add 3 zeros" to get 35,000 — what's really happening is the decimal point moves 3 places to the right.
Whole Number View
35
× 1,000
= 35,000
"Add 3 zeros to the end"
Decimal Point View
35.
→ → →
35,000.
Decimal moves 3 places RIGHT
Every whole number has a hidden decimal point at the end — that's the key!
🤝 Turn & Talk
💬Partner Discussion
If "adding zeros" and "moving the decimal" are the same strategy, why do we need BOTH ways of thinking about it?
Sentence starter: "I think both methods are useful because..."
Think about: When is "add zeros" easier? When is "move the decimal" easier?
🔄 Now Let's Go the Other Way!
We just learned: Multiplying by a power of 10 moves the decimal point to the RIGHT →
What about DIVIDING?
If multiplying makes numbers bigger (decimal moves right), then dividing should make numbers smaller...
← DIVIDE = Move Decimal LEFT
Multiply and divide are opposites — so the decimal moves in opposite directions!
👨🏫 Dividing Whole Numbers by Powers of 10
4,500
÷ 10
1
Find the hidden decimal point: 4,500.
2
÷ 10 = move decimal 1 place LEFT ←
3
4,500. → 450.0 → 450
4,500
÷ 10
= 450
Check: 450 × 10 = 4,500 ✓
👨🏫 More Division Examples
÷ 100 (2 places left)
7,200
÷ 100
= 72
7,200. → 72.00 → 72
÷ 1,000 (3 places left)
56,000
÷ 1,000
= 56
56,000. → 56.000 → 56
Using exponents: 7,200 ÷ 102 = 72 | 56,000 ÷ 103 = 56
👨🏫 Dividing Decimals by Powers of 10
Same rule — move the decimal point LEFT. But now we get smaller decimals!
1️⃣÷ 10
34.5
÷ 10
= 3.45
Decimal moves 1 place LEFT ←
2️⃣÷ 100
34.5
÷ 100
= 0.345
Decimal moves 2 places LEFT ←← (add leading zero)
3️⃣÷ 1,000
34.5
÷ 1,000
= 0.0345
Decimal moves 3 places LEFT ←←← (add placeholder zeros)
👨🏫 Division with Exponent Form
The exponent tells you how many places to move — same rule as multiplication, just go LEFT instead of RIGHT.
9.2
÷ 102
1
The exponent is 2 → move decimal 2 places LEFT
2
9.2 → 0.92 → 0.092
3
We needed an extra placeholder zero! 9.2 only has 1 digit before the decimal.
9.2
÷ 102
= 0.092
Check: 0.092 × 100 = 9.2 ✓
📌 Rule #2: DIVIDE = Move Decimal LEFT ←
← DIVIDE by a Power of 10 = Move Decimal LEFT
The exponent (or number of zeros) tells you how many places to move.
◀️÷ 10 or ÷ 101
Move 1 place LEFT
◀️◀️÷ 100 or ÷ 102
Move 2 places LEFT
◀️◀️◀️÷ 1,000 or ÷ 103
Move 3 places LEFT
📓 Write This Down
Key Word
Divide by Power of 10
In Your Notebook
DIVIDE by a power of 10 → move the decimal point LEFT ←
The exponent (or # of zeros) = how many places to move.
If you run out of digits, add placeholder zeros (e.g., 9.2 ÷ 100 = 0.092).
Memory trick: Divide = Decimal goes Down (Left on the number line).
✅ Quick Check
1️⃣6,300 ÷ 102
63
Move 2 places LEFT: 6,300. → 63.00
2️⃣5.7 ÷ 1,000
0.0057
Move 3 places LEFT: 5.7 → 0.57 → 0.057 → 0.0057
⚠️ Tricky Ones — Watch Out!
🚫Mistake #1: Slapping Zeros on Decimals
2.5
× 100
= 2.500
"Add 2 zeros" ONLY works on whole numbers. For decimals, move the decimal.
Correct: 2.5 × 100 = 250 ✓
🚫Mistake #2: Forgetting Placeholder Zeros
3.1
÷ 1,000
= 0.31
You need to move 3 places, but only have 1 digit left of the decimal. Add zeros!
Correct: 3.1 ÷ 1,000 = 0.0031 ✓
🚫Mistake #3: Wrong Direction!
4.2
÷ 10
= 42
Dividing makes numbers smaller, not bigger! LEFT ← not RIGHT →
Correct: 4.2 ÷ 10 = 0.42 ✓
🤝 Turn & Talk
💬Mistake Detectives
Your friend says: "12.4 × 10³ = 12.4000 because you just add three zeros!"
With your partner, explain:
1. What mistake did they make?
2. What is the correct answer?
3. How would you help them fix their thinking?
"Add zeros" is a shortcut for whole numbers only.
For decimals, always move the decimal point.
12.4 × 10³ = 12,400 (move 3 places RIGHT)
🗺️ The Complete Picture — Meet Mr. DL!
MULTIPLY → RIGHT
× Power of 10
Move decimal RIGHT →
Numbers get BIGGER
3.45 × 102 = 345
12 × 103 = 12,000
DIVIDE ← LEFT
÷ Power of 10
Move decimal ← LEFT
Numbers get SMALLER
345 ÷ 102 = 3.45
12 ÷ 103 = 0.012
Mr. DL says: Multiply = Right → | Divide = Left ← | The exponent = how many places!
📓 Write This Down
Key Words
Multiply = RIGHT
Divide = LEFT
Divide = LEFT
In Your Notebook — Draw This!
× Power of 10 → decimal moves RIGHT → (bigger)
÷ Power of 10 → decimal moves ← LEFT (smaller)
The exponent = # of places to move.
Memory trick — Mr. DL:
Multiply = Right → (top half)
Divide = Left ← (bottom half)
Draw the Mr. DL badge in your notebook!
📊 Patterns Table: One Number, All Operations
Watch what happens to 4.56 as we multiply and divide by powers of 10:
| Operation | Exponent Form | Result | What Happened |
|---|---|---|---|
| ÷ 1,000 | ÷ 103 | 0.00456 | ← 3 places |
| ÷ 100 | ÷ 102 | 0.0456 | ← 2 places |
| ÷ 10 | ÷ 101 | 0.456 | ← 1 place |
| START | × 100 = × 1 | 4.56 | No change |
| × 10 | × 101 | 45.6 | → 1 place |
| × 100 | × 102 | 456 | → 2 places |
| × 1,000 | × 103 | 4,560 | → 3 places |
Notice: dividing "undoes" multiplying — 45.6 × 10 → 456 and 456 ÷ 10 → 45.6
✅ Quick Check
True or False?
If I multiply a number by 10³ and then divide the result by 10³, I get back to my original number.
👍 Thumbs up for TRUE 👎 Thumbs down for FALSE
TRUE!
Multiply and divide by the same power of 10 are inverse operations — they undo each other.
Example: 2.5 × 10³ = 2,500 → 2,500 ÷ 10³ = 2.5 ✓
🔄 Ready to Practice!
What we've learned so far:
1
Powers of 10 can be written in exponent form (10³) or standard form (1,000)
2
Multiply = move decimal RIGHT → (numbers get bigger)
3
Divide = move decimal ← LEFT (numbers get smaller)
4
The exponent (or # of zeros) tells you how many places to move
Now let's practice together! 💪
👥 We Do: Multiply by Powers of 10
Let's solve these together. Decide: How many places? Which direction?
🅰️6.03 × 10²
Move 2 places RIGHT →
6.03 → 60.3 → 603
🅱️0.47 × 1,000
× 1,000 = 3 zeros = 3 places RIGHT →
0.47 → 4.7 → 47 → 470
🅲58 × 10⁴
Move 4 places RIGHT →
58. → 580 → 5,800 → 58,000 → 580,000
👥 We Do: Divide by Powers of 10
Same idea — but now we move LEFT ←. Watch for placeholder zeros!
🅰️820 ÷ 10²
Move 2 places LEFT ←
820. → 82.0 → 8.2
🅱️15.6 ÷ 1,000
÷ 1,000 = 3 places LEFT ←
15.6 → 1.56 → 0.156 → 0.0156
🅲3.9 ÷ 102
Move 2 places LEFT ←
3.9 → 0.39 → 0.039
Placeholder zero needed!
👥 We Do: Mixed — Multiply OR Divide?
First decide: is it × or ÷? Then decide the direction!
1️⃣7.25 × 10³
× = RIGHT → 3 places
7,250
2️⃣490 ÷ 10
÷ = LEFT ← 1 place
49
3️⃣0.08 × 10²
× = RIGHT → 2 places
8
4️⃣62.1 ÷ 10³
÷ = LEFT ← 3 places
0.0621
👥 We Do: Two Ways to Write It
These pairs mean the same thing. Solve both and verify they match!
Exponent Form
2.8 × 102 = ?
280
Move 2 right: 2.8 → 28 → 280
Standard Form
2.8 × 100 = ?
280
Count zeros (2) → move 2 right: 2.8 → 28 → 280
Both forms give the same answer because 102 IS 100. Same power of 10, different notation!
🤝 Turn & Talk
💬Check Your Partner
Quiz your partner with ONE problem — you pick!
1. Choose a starting number (any decimal)
2. Choose × or ÷
3. Choose a power of 10 (10, 100, or 1,000)
4. Have your partner solve it
5. Check their answer together!
Example: "What is 5.67 × 100?"
✅ Quick Check
Which is correct?
0.03 × 104 = ?
A) 0.0003 B) 300 C) 30 D) 3,000
B) 300
× = RIGHT → 4 places
0.03 → 0.3 → 3 → 30 → 300
🔍 You Try: Multiply Power-Up!
Solve each on your own. Remember: × = decimal moves RIGHT →
1️⃣9.04 × 10²
904
2️⃣0.006 × 1,000
6
3️⃣41.7 × 103
41,700
4️⃣0.52 × 104
5,200
🔍 You Try: Divide Power-Down!
Now the other direction. Remember: ÷ = decimal moves ← LEFT
1️⃣730 ÷ 10
73
2️⃣48.2 ÷ 10²
0.482
3️⃣6.1 ÷ 1,000
0.0061
4️⃣250,000 ÷ 104
25
🔍 You Try: Mixed — Think First!
Step 1: Is it × or ÷? Step 2: Which direction? Step 3: How many places?
1️⃣3.08 × 10³
× = RIGHT → 3 places
3,080
2️⃣91,000 ÷ 10³
÷ = LEFT ← 3 places
91
3️⃣0.7 × 105
× = RIGHT → 5 places
70,000
4️⃣425.9 ÷ 10²
÷ = LEFT ← 2 places
4.259
🔍 You Try: Complete the Patterns Table
Fill in each blank. Start with 7.2 — then apply each operation.
| ÷ 1,000 | ÷ 100 | ÷ 10 | START | × 10 | × 100 | × 1,000 |
|---|---|---|---|---|---|---|
| ? | ? | ? | 7.2 | ? | ? | ? |
| 0.0072 | 0.072 | 0.72 | 7.2 | 72 | 720 | 7,200 |
🏆 Challenge: Word Problem
A scientist measures a tiny organism that is 0.035 centimeters long. She needs to report its length in a journal that uses a unit 1,000 times smaller (micrometers). What number does she report?
Think Through It
- What is the starting number?
- Is the new unit smaller or bigger? (× or ÷?)
- What power of 10 is involved?
- Solve and write a complete sentence.
Converting to a smaller unit → the number gets BIGGER → multiply
0.035
× 1,000
= 35
The organism is 35 micrometers long.
🏆 Challenge: Word Problem
A factory produces 84,000 paper clips per day. The paper clips are packed into boxes of 10³ (1,000) each. How many boxes does the factory fill each day?
Think Through It
- What is the total? What's in each box?
- Are we splitting into groups? (× or ÷?)
- What power of 10 is 10³?
- Solve!
Splitting into equal groups → divide
84,000
÷ 10³
= 84
Move 3 places LEFT: 84,000 → 84. The factory fills 84 boxes each day.
🏆 Challenge: Think Backwards!
These problems give you the answer — find the missing piece!
🅰️___ × 10² = 450
Think: what ÷ 100 gives 450? Or: 450 ÷ 100 = 4.5
🅱️62.5 ÷ ___ = 0.625
Decimal moved 2 places LEFT → divided by 100 (or 10²)
🅲0.009 × 10? = 90
Decimal moved 4 places RIGHT → exponent = 4
(0.009 × 104 = 90)
📓 Summary Note
Key Words
Powers of 10
Multiply → RIGHT
Divide → LEFT
Exponent = # of places
Multiply → RIGHT
Divide → LEFT
Exponent = # of places
Write 1–2 Sentences
In the bottom of your notebook page, write 1–2 sentences explaining what you learned today about powers of 10.
Try to include:
• The word exponent
• The word decimal point
• The directions RIGHT and LEFT
Example: "When you multiply by a power of 10, the decimal point moves right. The exponent tells you how many places."
📝 Today's Key Takeaways
Big Idea #1
A power of 10 can be written two ways: exponent form (10³) or standard form (1,000). Both mean the same thing.
Big Idea #2
Multiply → RIGHT → (bigger numbers)
Divide → ← LEFT (smaller numbers)
The exponent or # of zeros = how many places.
Big Idea #3
"Adding zeros" (whole numbers) and "moving the decimal" are the same strategy. Every number has a hidden decimal point!
🎟️ Exit Ticket
1️⃣Solve: 5.03 × 10³
5,030
2️⃣Solve: 820 ÷ 10²
8.2
3️⃣True or False: 0.4 × 100 = 0.400
FALSE
0.4 × 100 = 40 (move decimal 2 places RIGHT, don't just add zeros to a decimal!)
4️⃣Fill in: 7.6 × 10? = 76,000
Decimal moved 4 places RIGHT → exponent = 4