Exponent / Power Calculator

Exponents (powers) represent repeated multiplication. The expression a^n means multiplying a by itself n times. This calculator handles positive, negative and fractional exponents as well as expressions involving multiple operations.

Negative exponents produce reciprocals: a^(−n) = 1/a^n. Fractional exponents represent roots: a^(1/2) is the square root of a, and a^(1/3) is the cube root. The calculator shows full step-by-step working so you can follow the logic.

What is an exponent? Repeated multiplication. 5² = 5 × 5 = 25. 2³ = 2 × 2 × 2 = 8. 10⁴ = 10,000. Notation: base^exponent. Special cases: any number to power 0 = 1 (by definition). Any number to power 1 = itself. 0^0 = undefined (or 1 by convention in many contexts). Negative exponents: 5⁻² = 1/5² = 1/25. Fractional exponents: 9^(1/2) = √9 = 3. UK GCSE Maths covers integer exponents; A-level adds fractional and irrational exponents.

Laws of exponents. Multiplication: a^m × a^n = a^(m+n). Sample: 2³ × 2⁴ = 2⁷ = 128. Division: a^m ÷ a^n = a^(m−n). Sample: 5⁵ ÷ 5² = 5³ = 125. Power of power: (a^m)^n = a^(m×n). Sample: (3²)³ = 3⁶ = 729. Distributive: (ab)^n = a^n × b^n. Negative: a^(−n) = 1/a^n. Zero: a⁰ = 1 (for a ≠ 0). Always taught in UK GCSE Higher Tier as 'index laws' or 'exponent laws'.

Scientific notation. Express very large or very small numbers compactly. 1,000,000 = 1 × 10⁶. 0.000001 = 1 × 10⁻⁶. UK distances: Earth to Moon 3.844 × 10⁸ m; Earth to Sun 1.496 × 10¹¹ m; Hydrogen atom radius 5.3 × 10⁻¹¹ m; mass of electron 9.109 × 10⁻³¹ kg; speed of light 2.998 × 10⁸ m/s. UK A-level Physics, Chemistry, Maths use scientific notation extensively. Calculator EXP or EE button enters scientific notation.

Powers of 2 and computer science. 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64, 2⁷ = 128, 2⁸ = 256, 2¹⁰ = 1,024, 2¹⁶ = 65,536, 2³² = 4.3 billion (32-bit max), 2⁶⁴ = 18 quintillion (64-bit max). Storage: kilobyte ≈ 2¹⁰ bytes = 1,024 B; megabyte 2²⁰; gigabyte 2³⁰; terabyte 2⁴⁰. RAM addressing: 32-bit systems max 4GB; 64-bit systems virtually unlimited. UK Computer Science GCSE/A-level requires familiarity.

Compound interest — real-world exponents. Future value = Principal × (1 + rate)^years. £10,000 at 5% for 30 years: £10,000 × 1.05³⁰ = £10,000 × 4.32 = £43,219. Same money at 7% for 30 years: 1.07³⁰ = 7.61 → £76,123. Just 2% extra annual return = 76% more wealth. Rule of 72: years to double = 72 ÷ rate. 5% → 14.4 years; 7% → 10.3; 10% → 7.2. Pension projections, mortgage calculations, investment forecasts — all use exponential growth.