Exponent Calculator

This exponent calculator raises any base b to any power n and shows the result instantly. It is sometimes called a power calculator, an "a to the power of b" calculator, or an index calculator. It handles every kind of exponent you are likely to meet: whole numbers (2³ = 8), zero (anything⁰ = 1), negative powers (2⁻³ = 0.125), decimals (1.07⁵ for compound growth), and fractional powers, which are really roots (9^1/2 = 3). It is useful for school and college students learning the laws of indices, for anyone doing scientific notation, and for everyday compound-growth sums such as interest, population, or area scaling.

What an exponent actually means

An exponent is just shorthand for repeated multiplication. In the expression bⁿ, the number b is the base and n is the exponent (also called the power or index). When n is a positive whole number, bⁿ = b × b × … × b, with the base written down n times. So 5⁴ = 5 × 5 × 5 × 5 = 625. The exponent counts the factors; it does not mean "multiply b by n" — a very common confusion. 5⁴ is 625, not 20.

The laws of exponents

These identities let you simplify powers without expanding everything by hand. They all follow directly from the "repeated multiplication" idea.

Product rule: b^m × bⁿ = b^(m+n). Same base, add the powers — e.g. 2³ × 2⁴ = 2⁷ = 128.

Quotient rule: b^m ÷ bⁿ = b^(m−n). Same base, subtract the powers.

Power of a power: (b^m)ⁿ = b^(m×n). Multiply the powers — e.g. (3²)³ = 3⁶ = 729.

Power of a product: (a × b)ⁿ = aⁿ × bⁿ.

Zero power: b⁰ = 1 for any base b ≠ 0. (The case 0⁰ is left undefined / indeterminate.)

Negative power: b⁽⁻ⁿ⁾ = 1 / bⁿ. A negative exponent means "take the reciprocal", so 2⁻³ = 1/2³ = 1/8 = 0.125. Because of this, raising 0 to a negative power is undefined (it would mean dividing by zero), and this calculator flags that case rather than printing a wrong number.

Fractional power: b^(1/n) = the nth root of b, written ⁿ√b. More generally b^(m/n) = ⁿ√(b^m) = (ⁿ√b)^m. So 8^(2/3) = (∛8)² = 2² = 4. Fractional exponents let you write any root as a power, which keeps algebra tidy.

Worked example

Suppose you invest a sum that grows 7% a year and you want the growth factor after 5 years. The multiplier each year is 1.07, so over 5 years the factor is 1.07⁵. Multiply step by step: 1.07² = 1.1449, then 1.07³ = 1.225043, 1.07⁴ ≈ 1.310796, and 1.07⁵ ≈ 1.402552. So the value grows by about 40.3% over five years. Type base 1.07 and exponent 5 above to confirm — this is the same maths behind compound-interest and SIP growth, just isolated to the power part.

Tips for using powers

Powers of 2 are worth memorising because they run through computing and data sizes: 2¹⁰ = 1,024 (about 1 KB), 2²⁰ ≈ 1 million, 2³⁰ ≈ 1 billion. For very large or very small results the calculator switches to scientific notation (for example 1.5 × 10⁹), so the digits stay readable. When you only need a root, remember you can enter a fractional exponent: 0.5 for a square root, 1/3 ≈ 0.3333 for a cube root.

Common mistakes

Watch the order of operations: without brackets, −3² means −(3²) = −9, while (−3)² = 9, because the power binds tighter than the leading minus sign. A negative base raised to an even power is positive and to an odd power is negative: (−2)³ = −8. Finally, a negative base with a non-integer exponent (such as (−4)^0.5) has no real value, since it asks for the square root of a negative number.