Exponent Calculator
Raise any base to any power — positive, negative, or fractional — or work backwards: give the result and solve for the missing base or the missing exponent. Steps are shown for every mode.
Example: with Solve for Result (bⁿ = ?) · Base (b) 2 · Exponent (n) 10 → Result (bⁿ): 1,024.
- Scientific notation1.0240 × 10^3
- Steps2^10 = 1,024
Computed by the calculator below using its default values. Change any input to see your own numbers.
Exponent rules in brief
An exponent counts repeated multiplication: 2ⁿ with n = 10 means ten 2s multiplied together, 1,024. A negative exponent flips the value into a reciprocal (2⁻³ = 1/8), a fractional exponent takes a root (9⁰⋅⁵ = √9 = 3), and any nonzero base to the 0 is 1. The product rule (bᵐ×bⁿ = bᵐ⁺ⁿ) and power rule ((bᵐ)ⁿ = bᵐⁿ) let you combine exponents instead of grinding out the multiplication.
How it’s calculated
Result mode computes bⁿ (odd roots of negative bases are handled as −|b|ⁿ when n = 1/k with k odd). Base mode inverts with a root: b = r^(1/n). Exponent mode uses logarithms: n = ln(r) ÷ ln(b), valid for positive b ≠ 1 and positive r. Values display to 10 significant digits.
A negative base with a non-integer exponent has no real answer — the calculator says so rather than returning a misleading number.
Common mistakes
- Reading 2⁻³ as −8 — a negative exponent means reciprocal (0.125), not a negative result.
- Writing −3⁴ when you mean (−3)⁴: without parentheses the minus applies after the power, giving −81 instead of 81.
- Multiplying base by exponent (2¹⁰ is 1,024, not 20).
- Adding exponents when bases differ — bᵐ×bⁿ = bᵐ⁺ⁿ only works for the same base.
Frequently asked questions
What does a negative exponent mean?
A negative exponent is a reciprocal: b^−n = 1 ÷ bⁿ. So 2^−3 = 1/2³ = 1/8 = 0.125. The base is not made negative — only inverted.
What does a fractional exponent mean?
A fractional exponent is a root: b^(1/n) is the nth root of b, and b^(m/n) is the nth root of b raised to the mth power. For example 9^0.5 = √9 = 3, and 8^(2/3) = (∛8)² = 4.
Can a negative base have a fractional exponent?
Not in the real numbers, in general — (−9)^0.5 would be the square root of a negative number, which is complex. Negative bases work fine with integer exponents ((−3)^4 = 81, (−3)^3 = −27), and odd roots of negatives are real, which the calculator handles when solving for the base.
How do I solve for an unknown exponent?
Use logarithms: if bⁿ = r then n = ln(r) ÷ ln(b). For 2ⁿ = 1,024, n = ln 1024 ÷ ln 2 = 10. Pick “Solve for exponent” mode and the calculator applies this change-of-base step for you.
What is anything to the power of zero?
Any nonzero base to the power 0 equals 1, because bⁿ ÷ bⁿ = b⁰ = 1. The expression 0⁰ is left undefined in most contexts (some fields define it as 1 by convention).