Exponent Calculator

Exponents are a shorthand for repeated multiplication. This calculator evaluates powers with any real base and exponent and can run the calculation backward using roots and logarithms.

What is an exponent?

In the expression bⁿ, b is the base and n is the exponent or power. When n is a positive whole number, bⁿ means b multiplied by itself n times: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. It is read as two to the fifth power, or two to the fifth. Powers of 2 and 3 have special names: b² is b squared and b³ is b cubed, after the area of a square and the volume of a cube.

Exponents grow quickly. Folding a sheet of paper doubles its thickness each time, so after 42 folds a 0.1 mm sheet would be 0.1 × 2⁴² mm thick, more than the distance to the Moon.

Basic exponent laws and rules

A handful of rules let you simplify and combine powers. They all follow from the definition as repeated multiplication, and they extend the meaning of exponents to zero, negative numbers and fractions in a consistent way.

Product rule. When multiplying powers with the same base, add the exponents: bᵐ × bⁿ = bᵐ⁺ⁿ. For example 2³ × 2⁴ = 2⁷ = 128, because three 2s times four 2s is seven 2s.

Quotient rule. When dividing powers with the same base, subtract the exponents: bᵐ ÷ bⁿ = bᵐ⁻ⁿ. So 5⁶ ÷ 5² = 5⁴ = 625.

Power of a power. Raising a power to another power multiplies the exponents: (bᵐ)ⁿ = bᵐⁿ. (3²)³ = 3⁶ = 729.

Power of a product and of a quotient. An exponent outside parentheses applies to every factor inside: (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ. So (2 × 5)³ = 8 × 125 = 1,000. There is no such rule for sums: (a + b)² is a² + 2ab + b², not a² + b².

Zero exponent. Any non-zero number raised to the power 0 equals 1. This follows from the quotient rule, since bⁿ ÷ bⁿ = b⁰ and any non-zero number divided by itself is 1. The expression 0⁰ is left undefined in many contexts, though it is taken as 1 in combinatorics and in most programming languages.

Negative exponents. A negative exponent means a reciprocal: b⁻ⁿ = 1 ÷ bⁿ. So 2⁻³ = 1/8 = 0.125, and 10⁻⁶ is one millionth. This is how scientific notation writes very small numbers.

Fractional exponents. An exponent of 1/n means the nth root: b^(1/2) = √b and b^(1/3) = ∛b. More generally b^(m/n) is the nth root of b raised to the m: 8^(2/3) = (∛8)² = 4. For a negative base, fractional exponents only give real answers when the root is odd, which is why this calculator reports (−8)^(1/3) = −2 but has no real value for (−8)^(1/2).

Negative bases. With a whole-number exponent, a negative base gives a positive result for even powers and a negative result for odd powers: (−2)⁴ = 16 and (−2)³ = −8. Note that −2⁴ means −(2⁴) = −16, since the exponent applies before the minus sign.

Solving for an unknown. If the result and exponent are known, take a root to find the base: if b⁵ = 32, then b = 32^(1/5) = 2. If the result and base are known, use logarithms to find the exponent: if 2ⁿ = 32, then n = log 32 ÷ log 2 = 5.

Frequently asked questions