Powers of 10: Patterns in Zeros and Decimals

In this lesson:

• Explain why multiplying by powers of 10 adds zeros to whole numbers
• Predict how the decimal point shifts when multiplying or dividing
• Use exponent notation to write powers of 10

What You Will Learn Today

1. Explain why multiplying whole numbers by powers of 10 appends zeros
2. Predict the zeros in any product
3. Explain why decimals shift when multiplied or divided
4. Compute products and quotients with powers of 10
5. Write powers of 10 using exponents
6. Connect exponents to shifts

Quick Review: Place Value and Times 10

From our last lesson (5.NBT.A.1):

• Moving one place left multiplies a digit's value by 10
• Moving one place right divides a digit's value by 10

Today's question: What happens when we multiply an entire number by 10, 100, or 1,000?

Digits Shift Left When Multiplied by 10

Each digit moves left. Zeros fill the empty places.

Example: Multiply 45 by Powers of 10

45 × 10: Both digits shift one place left

45 × 100: Both digits shift two places left

45 × 1,000: Both digits shift three places left

Why the Zeros Pattern Always Works

Each zero represents one factor of 10 — one leftward shift.

Quick Check: Predict the Product

Without computing, predict:

Think: How many zeros does 10,000 have? How many places will the digits shift?

Pause and predict before advancing...

Multiplying Decimals: Digits Still Shift Left

• Multiply 4.6 by 10: each digit shifts one place left
• The 4 moves from ones to tens; the 6 moves from tenths to ones
• Result: 46 (not 4.60)

The digits move left. The decimal point stays put.

Example: Multiply 0.35 by Powers of 10

0.35 × 10: Shift one place left

0.35 × 100: Shift two places left

0.35 × 1,000: Shift three places left

Dividing Reverses the Shift: Digits Move Right

Dividing by 10: Each digit shifts one place right

• 460 ÷ 10 = 46
• 46 ÷ 10 = 4.6
• 4.6 ÷ 10 = 0.46

Size check: Dividing makes numbers smaller. If your answer is bigger, the shift went the wrong way.

Watch Out: "Add Zeros" Fails for Decimals

A common mistake:

Why it's wrong: 3.40 equals 3.4 — adding a zero after the decimal does not change the value.

Correct reasoning: Shift digits one place left.

Quick Check: Multiply and Divide

Solve these mentally:

1. 0.07 × 100 = ?
2. 5.2 ÷ 10 = ?

Remember: Multiply → digits shift left. Divide → digits shift right.

Pause and solve before advancing...

Your Turn: Mixed Multiply and Divide

Compute each product or quotient:

1. 8 × 1,000 = ?
2. 0.06 × 100 = ?
3. 340 ÷ 100 = ?
4. 0.5 × 10 = ?

Pause and try each one before advancing...

Answers: Mixed Multiply and Divide Practice

1. 8 × 1,000 = 8,000 (3 places left)
2. 0.06 × 100 = 6 (2 places left)
3. 340 ÷ 100 = 3.4 (2 places right)
4. 0.5 × 10 = 5 (1 place left)

Exponent Notation for Powers of 10

The exponent tells how many times 10 is used as a factor.

Warning: Exponent Is Not Multiplication

These are completely different:

1,000 is not 30.

The exponent counts how many tens are multiplied together, not what 10 is multiplied by.

Example: Evaluate Expressions with Exponent Notation

Multiply:

Step 1: → shift 3 places left

Divide:

Step 1: → shift 2 places right

One Rule Unifies All Three Patterns

• Exponent = factors of 10
• Zeros = zeros in standard form
• Shift = places digits move

Multiply → shift left (bigger). Divide → shift right (smaller).

Quick Check: Exponent Notation Practice

Evaluate each expression:

1. = ?
2. = ?
3. = ?

Pause and solve before advancing...

Your Turn: Evaluate These Exponent Expressions

Evaluate each expression:

1. = ?
2. = ?
3. = ?
4. = ?

Pause and try each one before advancing...

Answers: Exponent Expressions with Shift Counts

1. = 9,000 (3 places left)
2. = 5 (2 places left)
3. = 320 (2 places left)
4. = 6 (3 places right)

Summary: Powers of 10 Shift Digits

• ✓ Multiply → digits shift left; zeros fill empty places
• ✓ Divide → digits shift right
• ✓ Exponent = number of places shifted

3.4 × 10 = 34, not 3.40
, not 30
Multiply = bigger; Divide = smaller

What Comes Next in Place Value

Next lesson: Reading and writing decimals to thousandths (5.NBT.A.3)

How today connects:

• Place value shifting helps check your decimal work
• Powers of 10 support estimation and verification
• Exponent notation returns in Grade 8 with scientific notation