Exponent / Power Calculator

Exponents, or powers, are shorthand for repeated multiplication. The expression a^n means multiplying a by itself n times, so 5² is 5 × 5 = 25, 2³ is 2 × 2 × 2 = 8, and 10⁴ is 10,000. The number being multiplied is the base and the small raised figure is the exponent. Two special cases are worth fixing early: anything raised to the power 1 is itself, and anything raised to the power 0 is 1 by definition, provided the base is not zero. The expression 0^0 is undefined, although many contexts take it as 1 by convention.

Negative and fractional exponents extend the same idea rather than replacing it. A negative exponent produces a reciprocal, so a^(−n) = 1/a^n and 5⁻² is 1/25. A fractional exponent is a root: a^(1/2) is the square root of a, which makes 9^(1/2) equal to √9 = 3, and a^(1/3) is the cube root. This calculator handles positive, negative and fractional exponents as well as expressions that mix several operations, and it shows full step-by-step working so you can follow the logic. UK GCSE Maths covers integer exponents, with fractional and irrational ones added at A-level.

The index laws are what let you simplify an expression before computing it. Multiplying powers of the same base adds the exponents, so 2³ × 2⁴ = 2⁷ = 128. Dividing subtracts them, giving 5⁵ ÷ 5² = 5³ = 125. Raising a power to another power multiplies them, so (3²)³ = 3⁶ = 729. A product inside a bracket distributes, meaning (ab)^n = a^n × b^n. Together with a^(−n) = 1/a^n and a⁰ = 1 for any non-zero a, that is the full set taught at GCSE Higher Tier under the name index laws.

Scientific notation uses powers of ten to write very large and very small numbers compactly. A million becomes 1 × 10⁶ and 0.000001 becomes 1 × 10⁻⁶. The convention earns its keep in science, where the Earth sits 3.844 × 10⁸ m from the Moon and 1.496 × 10¹¹ m from the Sun, a hydrogen atom has a radius of 5.3 × 10⁻¹¹ m, an electron weighs 9.109 × 10⁻³¹ kg, and light travels at 2.998 × 10⁸ m/s. A-level Physics, Chemistry and Maths all lean on it heavily, and handheld calculators enter it through the EXP or EE button.

Powers of two run right through computing. The sequence 2, 4, 8, 16, 32, 64, 128, 256 leads on to 2¹⁰ = 1,024 and 2¹⁶ = 65,536, and further out to 2³² at roughly 4.3 billion and 2⁶⁴ at about 18 quintillion. Storage sizes follow the same ladder, with a kilobyte at 2¹⁰ bytes, or 1,024 B, a megabyte at 2²⁰, a gigabyte at 2³⁰ and a terabyte at 2⁴⁰. Memory addressing does too, which is why 32-bit systems top out at 4GB of RAM while 64-bit systems are effectively unlimited. UK Computer Science at GCSE and A-level assumes familiarity with all of it.

Compound growth is where most people meet exponents in real life. Future value equals the principal multiplied by (1 + rate)^years, so £10,000 growing at 5% for 30 years becomes £10,000 × 1.05³⁰, a factor of 4.32, or £43,219. Shift the return to 7% over the same period and the factor is 7.61, giving £76,123. Two extra percentage points a year produce 76% more wealth, purely through compounding. The rule of 72 gives a quick sense of the pace, since dividing 72 by the rate gives the doubling time: 5% takes 14.4 years, 7% takes 10.3 and 10% takes 7.2. Pension projections, mortgage calculations and investment forecasts all run on this arithmetic.