Exponential notation, properties of exponents, powers of 10
Middle School Bridge • 6-8
What happens when you multiply a number by itself over and over? Writing 2 × 2 × 2 × 2 × 2 gets tedious fast. Exponential notation gives us a compact way to express repeated multiplication. Beyond convenience, exponents unlock powerful patterns, especially with powers of 10 that underlie our entire number system.
An exponent tells you how many times to use the base as a factor.
Here, b is the base and n is the exponent (or power). For example, 34 = 3 × 3 × 3 × 3 = 81.
| Expression | Meaning | Value |
|---|---|---|
| 25 | 2 × 2 × 2 × 2 × 2 | 32 |
| 53 | 5 × 5 × 5 | 125 |
| 104 | 10 × 10 × 10 × 10 | 10,000 |
| 71 | 7 | 7 |
Evaluate (-2)4.
Notice: an even exponent on a negative base gives a positive result.
These rules let you work with exponents without expanding everything:
| Property | Rule | Example |
|---|---|---|
| Product Rule | am × an = am+n | 23 × 24 = 27 = 128 |
| Quotient Rule | am ÷ an = am-n | 56 ÷ 52 = 54 = 625 |
| Power of a Power | (am)n = amn | (32)3 = 36 = 729 |
Simplify: 43 × 42.
Same base (4), so add exponents: 43+2 = 45 = 1024.
Powers of 10 create the foundation of our place value system.
101 = 10, 102 = 100, 103 = 1,000, 106 = 1,000,000. The exponent tells you how many zeros follow the 1. This is why scientific notation uses powers of 10 to express very large or very small numbers.
Do not multiply the base by the exponent. 34 does NOT mean 3 × 4 = 12. It means 3 × 3 × 3 × 3 = 81. The exponent tells you how many times the base is used as a factor, not what to multiply the base by.
1. Evaluate 63.
6 × 6 × 6 = 216.
2. Evaluate (-3)3.
(-3) × (-3) × (-3) = 9 × (-3) = -27. An odd exponent on a negative base gives a negative result.
3. Simplify: 72 × 75.
Same base, add exponents: 77.
4. Simplify: 108 ÷ 103.
Subtract exponents: 105 = 100,000.
5. What is 20 + 50?
Any nonzero number to the 0th power is 1. So 20 + 50 = 1 + 1 = 2.
Exponents express repeated multiplication compactly. The base is the number being multiplied, and the exponent tells how many times. Key properties include the product rule (add exponents when multiplying same bases), quotient rule (subtract exponents when dividing), and power of a power rule (multiply exponents). Any nonzero number raised to the 0th power equals 1.