Powers of 10
Powers of 10 are at the heart of our decimal number system. Every place value position is a power of 10, which is why multiplying or dividing by 10 simply shifts digits one position left or right.
What Is a Power of 10?
A power of 10 means 10 multiplied by itself a certain number of times. We write this using an exponent: 10ⁿ means 10 multiplied n times.
| Power | Meaning | Value | Place Name |
|---|---|---|---|
| 10⁰ | 1 | 1 | Ones |
| 10¹ | 10 | 10 | Tens |
| 10² | 10 × 10 | 100 | Hundreds |
| 10³ | 10 × 10 × 10 | 1,000 | Thousands |
| 10⁴ | 10⁴ | 10,000 | Ten Thousands |
| 10⁵ | 10⁵ | 100,000 | Hundred Thousands |
| 10⁶ | 10⁶ | 1,000,000 | Millions |
Quick memory trick: 10ⁿ = 1 followed by n zeros.
Multiplying by Powers of 10
Multiplying a number by 10ⁿ moves every digit n places to the left.
Dividing by Powers of 10
Dividing by 10ⁿ moves every digit n places to the right.
Negative Powers of 10 (Decimals)
| Power | Value | Place Name |
|---|---|---|
| 10⁻¹ | 0.1 | Tenths |
| 10⁻² | 0.01 | Hundredths |
| 10⁻³ | 0.001 | Thousandths |
Pattern in Powers of 10
Each step multiplies by 10; each step back divides by 10. This consistent pattern is what makes our number system so powerful.
- 10ⁿ = 1 followed by n zeros.
- Every place value is a power of 10.
- Multiplying by 10ⁿ: shift digits n places left (or add n zeros).
- Dividing by 10ⁿ: shift digits n places right.
- Negative exponents give decimal place values (tenths, hundredths, …).
Quick Practice
- What is 10⁵?
- Calculate 32 × 10³.
- Calculate 45,000 ÷ 10².
- What power of 10 equals one million?
- Write 7 × 10⁻² as a decimal.
Summary
Powers of 10 are the backbone of the decimal number system. Every place column is exactly 10ⁿ for some integer n. Understanding this lets you mentally multiply or divide any number by powers of 10 in seconds, and lays the groundwork for scientific notation and logarithms.