Understanding Exponents
Exponentiation is a mathematical operation involving two numbers: the base (x) and the exponent (n). When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, xⁿ is the product of multiplying x by itself n times.
Core Concepts
- Positive Exponents: 2³ = 2 × 2 × 2 = 8
- Negative Exponents: Represent division or the reciprocal. 2⁻³ = 1/2³ = 1/8
- Zero Exponent: Any non-zero number to the power of 0 is 1. x⁰ = 1
- Fractional Exponents: Relate to roots. x^(½) = √x
Rules of Exponents
These fundamental laws allow you to simplify complex exponential expressions easily.
Product Rule
a^m × a^n = a^{m+n}
When multiplying terms with the same base, add their exponents.
Quotient Rule
a^m / a^n = a^{m-n}
When dividing terms with the same base, subtract the exponents.
Power Rule
(a^m)^n = a^{m×n}
When raising a power to another power, multiply the exponents.
Power of a Product
(ab)^n = a^n × b^n
The exponent distributes to each factor of the base.
Reference Table: Powers of 2 & 10
| Power (n) | 2^n | 10^n |
|---|---|---|
| 1 | 2 | 10 |
| 2 | 4 | 100 |
| 3 | 8 | 1,000 |
| 4 | 16 | 10,000 |
| 5 | 32 | 100,000 |
| 6 | 64 | 1,000,000 |
| 10 | 1,024 | 10,000,000,000 |