Exponent Calculator

Raise any real base to any real power. The calculator handles positive, negative, zero and fractional exponents, and shows the resulting value plus the step-by-step working. It also warns you when a fractional power of a negative base would produce a complex (non-real) result.

Enter base and exponent

Any real number: positive, negative, zero or fractional (e.g. 0.5 for square root).

Enter a base and an exponent, then press Calculate.

What an exponent means

An exponent tells you how many times to multiply a number by itself. In b^n, b is the base and n is the exponent (or power).

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

When n is a positive integer, b^n is just repeated multiplication. But the definition extends to every real number:

ExponentMeaningExample
Positive integerRepeated multiplication2³ = 8
ZeroResult is 1 (for any b ≠ 0)5⁰ = 1
Negative integerReciprocal of the positive power2⁻³ = 1/8 = 0.125
Fraction 1/nn-th root9^(1/2) = 3
Fraction p/qq-th root, then raise to p8^(2/3) = 4
Real numberDefined via exp / log for b > 02^π ≈ 8.8249…

Exponent rules

  • Product rule: bᵐ × bⁿ = b^(m+n)
  • Quotient rule: bᵐ ÷ bⁿ = b^(m−n) (b ≠ 0)
  • Power of a power: (bᵐ)ⁿ = b^(m×n)
  • Power of a product: (a·b)ⁿ = aⁿ·bⁿ
  • Negative exponent: b⁻ⁿ = 1 / bⁿ
  • Zero exponent: b⁰ = 1 (b ≠ 0)
  • Fractional exponent: b^(1/n) = ⁿ√b

Worked examples

Example 1 — Positive integer exponent

Calculate 3⁴.

3 × 3 × 3 × 3 = 81.

3⁴ = 81

Example 2 — Negative exponent

Calculate 5⁻².

5⁻² = 1 / 5² = 1 / 25 = 0.04.

5⁻² = 0.04

Example 3 — Fractional exponent

Calculate 27^(1/3) (cube root of 27).

3 × 3 × 3 = 27, so 27^(1/3) = 3.

27^(1/3) = 3

Example 4 — Fractional exponent with numerator > 1

Calculate 16^(3/4).

16^(1/4) = 2 (fourth root), then 2³ = 8.

16^(3/4) = 8

Common mistakes

  • Confusing bⁿ with b × n. 2⁵ = 32, not 10. Exponents mean repeated multiplication, not multiplication of base by exponent.
  • Multiplying exponents when adding bases. bᵐ + bⁿ cannot be simplified. Only the product and quotient rules let you combine exponents.
  • Forgetting that b⁰ = 1 for all b ≠ 0. Yes, even 1000⁰ = 1 and (−5)⁰ = 1.
  • Assuming 0⁰ = 1. 0⁰ is undefined in most contexts; this calculator returns an error rather than a number.
  • Taking fractional powers of negative bases. (−8)^(1/3) = −2 is real because 3 is odd, but (−4)^(1/2) is not a real number. The calculator warns when the result falls outside the real numbers.

Square Root Calculator

Compute square roots with simplification.

Cube Root Calculator

Compute cube roots including negative numbers.

Logarithm Calculator

The inverse of exponentiation.

Scientific Calculator

Full scientific calculator with trig and log.

Frequently asked questions

b⁻ⁿ = 1 / bⁿ. For example, 2⁻³ = 1 / 2³ = 1/8. Negative exponents do not produce negative numbers — they produce reciprocals.

b^(1/n) is the n-th root of b. So 9^(1/2) = 3 (square root), 27^(1/3) = 3 (cube root), and 16^(1/4) = 2 (fourth root). A general fraction p/q means: take the q-th root first, then raise to the p-th power.

For any b ≠ 0, b⁰ = 1. This follows from the quotient rule: b^n / b^n = b^(n−n) = b⁰ and any non-zero number divided by itself is 1. 0⁰ is undefined.

A real cube root exists for negative numbers because (−2)³ = −8. But there is no real number whose square is negative — (−2)² = 4, not −4. Even roots of negative numbers produce complex results, which this calculator flags as an error.