Exponent Calculator
Raise any base to any power — negative, fractional or huge — with the expanded multiplication, scientific notation and the rule that applies shown.
Exponent Calculator: with the default inputs, result is 1,024.
The number being multiplied.
How many times to multiply. Negatives and fractions are fine.
- Scientific notation
- 1.024 × 10³
- Expanded form
- 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
- Reciprocal (base⁻ᵉˣᵖ)
- 0.0009765625
Assumptions
- 0⁰ is returned as 1 (with a warning).
- Fractional exponents are matched to a fraction p/q with q ≤ 64 for the step-by-step root form; otherwise the result is computed with logarithms.
How this is worked out
The formula
bⁿ = b × b × … × b (n factors) b⁰ = 1 (b ≠ 0) b⁻ⁿ = 1 ÷ bⁿ b^(p/q) = (ᵠ√b)ᵖ bᵐ × bⁿ = bᵐ⁺ⁿ, (bᵐ)ⁿ = bᵐⁿ, bᵐ ÷ bⁿ = bᵐ⁻ⁿ
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Base
- The number being multiplied.defaults to 2
- Exponent (power)
- How many times to multiply. Negatives and fractions are fine.defaults to 10
What you get back
- Resultmain answer
- Scientific notation
- Expanded form
- Reciprocal (base⁻ᵉˣᵖ)
What this assumes
- 0⁰ is returned as 1 (with a warning).
- Fractional exponents are matched to a fraction p/q with q ≤ 64 for the step-by-step root form; otherwise the result is computed with logarithms.
About this calculator
An exponent is shorthand for repeated multiplication: 2¹⁰ means ten 2s multiplied together, 1,024. That's the whole idea, and every other exponent rule is what you get by insisting the shorthand stays consistent when the exponent is zero, negative, or a fraction.
How to use it
Enter a base and an exponent. The result appears with the expanded multiplication (for small whole-number powers), scientific notation for very large or small answers, and the reciprocal. Use Solve for to go backwards — "2 to what power is 1,000,000?" — which is really a logarithm.
Where the rules come from
- Zero exponent. Each time you lower the exponent by 1 you divide by the base: 2³ = 8, 2² = 4, 2¹ = 2, so 2⁰ must be 1. This is why anything (except 0) to the zero power is 1.
- Negative exponents. Keep dividing: 2⁻¹ = ½, 2⁻² = ¼. A negative exponent means "one over the positive power", never "a negative number".
- Fractional exponents. If 8^(1/3) × 8^(1/3) × 8^(1/3) has to equal 8¹, then 8^(1/3) must be the cube root of 8, which is 2. So b^(p/q) is the q-th root of b, raised to the p-th power — take the root first to keep the numbers small.
- Negative bases. (−2)⁴ = 16 but (−2)³ = −8: even powers cancel the sign, odd powers keep it. Note that −2⁴ without parentheses means −(2⁴) = −16 — the exponent binds before the minus sign.
Limits
A negative base with a fractional exponent — (−8)^(1/2), say — has no real answer; the calculator says so rather than inventing one. Results above about 1.8 × 10³⁰⁸ overflow double-precision arithmetic; the error message gives log₁₀ of the answer instead, which is usually what you actually need at that scale.
Common mistakes
- Multiplying base by exponent (2³ is 8, not 6).
- Treating 2⁻³ as −8 (it's 1/8).
- Adding exponents when the bases differ: 2³ × 3² is not 6⁵.
Frequently asked questions
▸What does a negative exponent mean?
One divided by the positive power. 5⁻² = 1 ÷ 5² = 1/25 = 0.04. It does not make the answer negative.
▸What is any number to the power of 0?
1, for any non-zero base. 0⁰ is left undefined in analysis, though many calculators and programming languages return 1.
▸How do I calculate a fractional exponent by hand?
Write the exponent as p/q, take the q-th root of the base, then raise it to the p-th power. 27^(2/3) = (∛27)² = 3² = 9.
▸Why is (−8)^(1/3) an error when the cube root of −8 is −2?
In the real numbers −2 is a valid cube root of −8, but exponent rules for fractional powers only stay consistent for non-negative bases, so the principal value is complex. For odd roots of negative numbers use the Square Root Calculator with degree 3.
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