What an exponent means
An exponent says how many times to multiply a number by itself. 25 = 2 × 2 × 2 × 2 × 2 = 32. The number being multiplied (2) is the base and the small raised number (5) is the exponent or power.
The exponent rules
| Rule | In symbols | Example |
|---|---|---|
| Multiplying the same base: add the exponents | bᵐ × bⁿ = bᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Dividing the same base: subtract the exponents | bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 5⁶ ÷ 5⁴ = 5² = 25 |
| A power of a power: multiply the exponents | (bᵐ)ⁿ = bᵐⁿ | (3²)³ = 3⁶ = 729 |
| A product in brackets: each part gets the power | (ab)ⁿ = aⁿbⁿ | (2 × 5)³ = 8 × 125 = 1,000 |
| Power of zero | b⁰ = 1 | 7⁰ = 1 |
| Negative exponent: one over the positive power | b⁻ⁿ = 1 ÷ bⁿ | 2⁻³ = 1/8 |
| Fractional exponent: a root | b^(m/n) = (ⁿ√b)ᵐ | 8^(2/3) = (∛8)² = 4 |
So when the exponents are in brackets like (x³)², you multiply them: x⁶. When the same base is multiplied, like x³ × x², you add them: x⁵.
Big powers, exactly
Ordinary calculators round big powers. Calcist works out whole-number powers exactly, so 2100 shows all 31 digits: 1,267,650,600,228,229,401,496,703,205,376.
Finding a missing exponent or base
To find the exponent in 2ˣ = 1024, use logarithms: x = log(1024) ÷ log(2) = 10. To find the base in b³ = 125, take the cube root: b = 5.