Multiply Decimals by Powers of Ten

Fifth Grade Math - Complete Guide

📚 What are Powers of Ten?

Definition

Powers of ten are numbers like 10, 100, 1000, etc. They are 10 multiplied by itself a certain number of times.

Standard Form

\(10 = 10\) (1 zero)
\(100 = 10 \times 10\) (2 zeros)
\(1000 = 10 \times 10 \times 10\) (3 zeros)
\(10000 = 10 \times 10 \times 10 \times 10\) (4 zeros)

Exponential Form (Using Exponents)

\(10^1 = 10\)
\(10^2 = 100\)
\(10^3 = 1000\)
\(10^4 = 10000\)

💡 Key Pattern

Important Rule:

The number of zeros in the power of ten = The exponent
Example: \(10^5\) has 5 zeros → \(100000\)

✖️ Multiply a Decimal by a Power of Ten

The Golden Rule

✅ Move the decimal point to the RIGHT

Number of places = Number of zeros in the power of ten

Steps:

Count the zeros in the power of ten
Move the decimal point to the right that many places
Add zeros if needed

💡 Examples

Example 1: \(3.2 \times 10\)

Step 1: Count zeros in 10 → 1 zero
Step 2: Move decimal point right 1 place
Step 3: \(3.2 \rightarrow 32.\) or just \(32\)

✓ Answer: \(3.2 \times 10 = 32\)

Example 2: \(4.56 \times 100\)

Step 1: Count zeros in 100 → 2 zeros
Step 2: Move decimal point right 2 places
Step 3: \(4.56 \rightarrow 456.\) or just \(456\)

✓ Answer: \(4.56 \times 100 = 456\)

Example 3: \(0.75 \times 1000\)

Step 1: Count zeros in 1000 → 3 zeros
Step 2: Move decimal point right 3 places
Step 3: \(0.75 \rightarrow 0.750. \rightarrow 750.\) (add 1 zero, then move)

✓ Answer: \(0.75 \times 1000 = 750\)

🔢 Multiply by Powers of Ten with Exponents

Understanding Exponents

When you see \(10^n\), the exponent \(n\) tells you how many places to move the decimal point.

✅ The Exponent Rule

Exponent value = Number of places to move decimal RIGHT

Common Powers of Ten

Exponential Form	Standard Form	Places to Move
\(10^1\)	10	1
\(10^2\)	100	2
\(10^3\)	1000	3
\(10^4\)	10000	4

💡 Examples

Example 1: \(5.6 \times 10^2\)

Step 1: Look at exponent → 2
Step 2: Move decimal right 2 places
Step 3: \(5.6 \rightarrow 5.60 \rightarrow 560.\) (add zero, then move)

✓ Answer: \(5.6 \times 10^2 = 560\)

Example 2: \(0.084 \times 10^3\)

Step 1: Look at exponent → 3
Step 2: Move decimal right 3 places
Step 3: \(0.084 \rightarrow 084. = 84\)

✓ Answer: \(0.084 \times 10^3 = 84\)

Example 3: \(1.2345 \times 10^4\)

Step 1: Look at exponent → 4
Step 2: Move decimal right 4 places
Step 3: \(1.2345 \rightarrow 12345.\) = \(12345\)

✓ Answer: \(1.2345 \times 10^4 = 12345\)

🔻 Multiply by 0.1 or 0.01

The Opposite Rule!

When multiplying by 0.1, 0.01, or 0.001, we move the decimal point to the LEFT (opposite direction!).

⚠️ Move the decimal point to the LEFT

Number of places = Number of zeros in 0.1, 0.01, or 0.001

Understanding the Pattern

Multiplier	Zeros	Places to Move LEFT	Same as
\(0.1\)	1	1	÷ 10
\(0.01\)	2	2	÷ 100
\(0.001\)	3	3	÷ 1000

💡 Important Note:

Multiplying by 0.1 is the SAME as dividing by 10!
Multiplying by 0.01 is the SAME as dividing by 100!

💡 Examples

Example 1: \(45 \times 0.1\)

Step 1: Count zeros in 0.1 → 1 zero
Step 2: Move decimal LEFT 1 place
Step 3: \(45. \rightarrow 4.5\)

✓ Answer: \(45 \times 0.1 = 4.5\) (same as \(45 \div 10\))

Example 2: \(3.25 \times 0.01\)

Step 1: Count zeros in 0.01 → 2 zeros
Step 2: Move decimal LEFT 2 places
Step 3: \(3.25 \rightarrow 0.0325\) (add one zero)

✓ Answer: \(3.25 \times 0.01 = 0.0325\)

Example 3: \(876 \times 0.001\)

Step 1: Count zeros in 0.001 → 3 zeros
Step 2: Move decimal LEFT 3 places
Step 3: \(876. \rightarrow 0.876\)

✓ Answer: \(876 \times 0.001 = 0.876\)

🔍 Find the Missing Number

Strategy to Solve

When you have an equation like \(? \times 10 = 350\) or \(4.5 \times ? = 4500\), follow these steps:

Method: Count Decimal Places

Compare the two numbers
Count how many places the decimal moved
Determine the direction (left or right)
Find the power of ten that matches

✅ Remember:

• Decimal moved RIGHT → multiply by 10, 100, 1000, etc.
• Decimal moved LEFT → multiply by 0.1, 0.01, 0.001, etc.

💡 Examples

Example 1: \(3.5 \times ? = 350\)

Step 1: Compare → \(3.5\) to \(350\)
Step 2: Count decimal movement → 2 places to the RIGHT
Step 3: 2 places right = multiply by 100

✓ Answer: The missing number is \(100\)

Example 2: \(12 \times ? = 1.2\)

Step 1: Compare → \(12\) to \(1.2\)
Step 2: Count decimal movement → 1 place to the LEFT
Step 3: 1 place left = multiply by 0.1

✓ Answer: The missing number is \(0.1\)

Example 3: \(? \times 10^3 = 4560\)

Step 1: \(10^3 = 1000\) means multiply by 1000
Step 2: To find original: \(4560 \div 1000\)
Step 3: Move decimal 3 places LEFT → \(4.560\)

✓ Answer: The missing number is \(4.56\)

Example 4: \(0.625 \times ? = 6.25\)

Step 1: Compare → \(0.625\) to \(6.25\)
Step 2: Count decimal movement → 1 place to the RIGHT
Step 3: 1 place right = multiply by 10

✓ Answer: The missing number is \(10\) or \(10^1\)

📊 Visual Summary: Direction Guide

When You Multiply By...	Move Decimal...	Example
\(10\) or \(10^1\)	1 place RIGHT →	\(2.5 \times 10 = 25\)
\(100\) or \(10^2\)	2 places RIGHT →	\(3.7 \times 100 = 370\)
\(1000\) or \(10^3\)	3 places RIGHT →	\(0.45 \times 1000 = 450\)
\(0.1\)	1 place LEFT ←	\(25 \times 0.1 = 2.5\)
\(0.01\)	2 places LEFT ←	\(370 \times 0.01 = 3.7\)
\(0.001\)	3 places LEFT ←	\(450 \times 0.001 = 0.45\)