Exponents

An exponent tells you how many times a number is multiplied by itself.

Basic form: aⁿ

• a = base
• n = exponent

Read as: “a to the power of n”.

Example: 5³

This does NOT mean 5 × 3.

It means:

5 × 5 × 5 = 125

The exponent tells you how many times to use the base as a factor.

Exponent of 1

a¹ = a

Example: 7¹ = 7

Exponent of 0

a⁰ = 1   (a ≠ 0)

Example: 5⁰ = 1

Note: Anything (except 0) raised to the 0 power equals 1.

A negative exponent means take the reciprocal.

a^-n = 1 / aⁿ

Example:

2^-3 = 1 / 2³ = 1 / 8

Rule 1: Multiply Same Bases

a^m × aⁿ = a^m+n

Example: 2³ × 2⁴ = 2⁷

Rule 2: Divide Same Bases

a^m ÷ aⁿ = a^m-n

Example: 5⁶ ÷ 5² = 5⁴

Rule 3: Power of a Power

(a^m)ⁿ = a^m×n

Example: (3²)⁴ = 3⁸

Rule 4: Power of a Product

(ab)ⁿ = aⁿbⁿ

Example: (2×3)² = 2² × 3² = 4 × 9 = 36

Rule 5: Power of a Quotient

(a/b)ⁿ = aⁿ / bⁿ

• Mistake: 3² = 6
  Correct: 3² = 9
• Mistake: 2³ + 2³ = 2⁶
  Correct: 8 + 8 = 16
• Mistake: (2+3)² = 2² + 3²
  Correct: 5² = 25

Powers of 10 (Science & Math)

• 10² = 100
• 10⁶ = 1,000,000

Computer Science Connections

• 2⁸ = 256 (1 byte)
• 2¹⁰ = 1024 (about 1 KB)