Powers of Ten
Grade 5 Math - Complete Reference Guide
1. Understanding Powers of Ten
What are Powers of Ten?
Powers of ten are numbers created by multiplying 10 by itself a certain number of times. We use exponents to write powers of ten in a shorter way.
Key Terms:
📌 Base: The number being multiplied (always 10 in powers of ten)
📌 Exponent: The small number written above the base that shows how many times to multiply
📌 Power: The result of multiplying the base by itself
Three Ways to Write Powers of Ten:
| Exponential Form | Expanded Form | Standard Form |
|---|---|---|
| 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 |
The Pattern in Powers of Ten:
🔍 Pattern Discovery:
The exponent tells you how many zeros come after the 1!
- 101 = 10 → 1 zero
- 102 = 100 → 2 zeros
- 103 = 1,000 → 3 zeros
- 104 = 10,000 → 4 zeros
- 105 = 100,000 → 5 zeros
✨ Special Case: 100 = 1
Any number (except 0) raised to the power of 0 equals 1.
2. Evaluate Powers of Ten
What Does "Evaluate" Mean?
To evaluate means to find the value or calculate the result. When we evaluate powers of ten, we find what number the power of ten equals.
Formula to Remember:
10n = 1 followed by n zeros
(where n is the exponent)
Step-by-Step Method:
- Step 1: Look at the exponent (the small number)
- Step 2: Write the number 1
- Step 3: Add that many zeros after the 1
- Step 4: Add commas for easier reading (every 3 digits from right)
Examples:
Example 1: Evaluate 103
Step 1: The exponent is 3
Step 2: Write 1
Step 3: Add 3 zeros: 1000
Step 4: Add comma: 1,000
Answer: 103 = 1,000
Example 2: Evaluate 107
Step 1: The exponent is 7
Step 2: Write 1
Step 3: Add 7 zeros: 10000000
Step 4: Add commas: 10,000,000
Answer: 107 = 10,000,000 (ten million)
Example 3: Evaluate 105
The exponent is 5, so we need 1 followed by 5 zeros.
Answer: 105 = 100,000 (one hundred thousand)
💡 Quick Tip:
The exponent tells you exactly how many zeros to write! It's that simple!
3. Write Powers of Ten with Exponents
Going Backwards: From Standard Form to Exponential Form
Sometimes we need to write a large number using exponents. This makes very large numbers easier to write and read!
Steps to Write with Exponents:
- Step 1: Count the number of zeros in the number
- Step 2: Write 10 as the base
- Step 3: Use the number of zeros as the exponent
- Step 4: Write the exponent as a small number above the 10
Formula:
Number of zeros = Exponent
If there are n zeros after 1, write it as 10n
Examples:
Example 1: Write 1,000 using an exponent
Step 1: Count the zeros → 1,000 has 3 zeros
Step 2: Base is 10
Step 3: Exponent is 3
Answer: 1,000 = 103
Example 2: Write 1,000,000 using an exponent
Step 1: Count the zeros → 1,000,000 has 6 zeros
Step 2: Base is 10
Step 3: Exponent is 6
Answer: 1,000,000 = 106
Example 3: Write 10,000,000,000 using an exponent
Count the zeros → 10,000,000,000 has 10 zeros
Answer: 10,000,000,000 = 1010 (ten billion)
| Standard Form | Number of Zeros | Exponential Form | Name |
|---|---|---|---|
| 10 | 1 | 101 | Ten |
| 100 | 2 | 102 | Hundred |
| 1,000 | 3 | 103 | Thousand |
| 10,000 | 4 | 104 | Ten Thousand |
| 100,000 | 5 | 105 | Hundred Thousand |
| 1,000,000 | 6 | 106 | Million |
| 10,000,000 | 7 | 107 | Ten Million |
| 100,000,000 | 8 | 108 | Hundred Million |
| 1,000,000,000 | 9 | 109 | Billion |
4. How to Read Powers of Ten
Two Ways to Read Exponents:
| Exponential Form | How to Read It |
|---|---|
| 101 | "Ten to the first power" OR "Ten to the power of one" |
| 102 | "Ten to the second power" OR "Ten squared" |
| 103 | "Ten to the third power" OR "Ten cubed" |
| 104 | "Ten to the fourth power" OR "Ten to the power of four" |
| 105 | "Ten to the fifth power" OR "Ten to the power of five" |
| 106 | "Ten to the sixth power" OR "Ten to the power of six" |
5. Practice Problems
Type 1: Evaluate Powers
Problem: What is 104?
Solution: The exponent is 4, so write 1 followed by 4 zeros.
Answer: 10,000
Problem: What is 108?
Solution: The exponent is 8, so write 1 followed by 8 zeros.
Answer: 100,000,000
Type 2: Write with Exponents
Problem: Write 100,000 using an exponent.
Solution: Count the zeros. There are 5 zeros after the 1.
Answer: 105
Problem: Write 1,000,000,000 using an exponent.
Solution: Count the zeros. There are 9 zeros after the 1.
Answer: 109
Type 3: Compare Powers
Problem: Which is greater: 105 or 103?
Solution:
105 = 100,000
103 = 1,000
Answer: 105 is greater because 100,000 > 1,000
6. Real-Life Uses of Powers of Ten
Why Do We Use Powers of Ten?
Powers of ten help us write very large numbers in a shorter, easier way. This is especially useful in science, technology, and everyday life!
Real-Life Examples:
🌍 Population
The world population is about 8,000,000,000 people.
We can write this as: 8 × 109 people
💰 Money
A million dollars = $1,000,000 = $106
A billion dollars = $1,000,000,000 = $109
📏 Measurement
1 kilometer = 1,000 meters = 103 meters
1 centimeter = 0.01 meters = 10-2 meters
💻 Technology
1 gigabyte (GB) = 1,000,000,000 bytes = 109 bytes
Computer speeds are measured in billions of operations per second!
Quick Reference Summary
| Concept | Key Points |
|---|---|
| Base | The number being multiplied (always 10 in powers of ten) |
| Exponent | The small number above the base; shows how many times to multiply |
| Pattern | Number of zeros = Exponent |
| Evaluate 10n | Write 1 followed by n zeros |
| Write as Exponent | Count zeros, use that number as exponent |
| Special Case | 100 = 1 |
| Reading | 105 = "Ten to the fifth power" or "Ten to the power of five" |
To Evaluate
Exponent → Zeros
To Write
Zeros → Exponent
Remember
Count carefully!
📚 Study Tips:
- The exponent always equals the number of zeros!
- Practice counting zeros carefully
- Remember: 100 = 1 and 101 = 10
- Use powers of ten to make large numbers easier to write
- The pattern works both ways: evaluate or write with exponents