Enter a base and an exponent and this calculator returns the power immediately, along with the expanded multiplication for small whole exponents and scientific notation once results get large. Switch the mode dropdown to work backwards instead — give it a base and a result to recover the exponent (a logarithm), or an exponent and a result to recover the base (a root). For plain roots, our square root calculator also gives simplified radical form, and the percentage calculator handles growth problems that would otherwise need powers.
Exponent Calculator
Any base to any power — negative, fractional or decimal — with the working shown and a solver for the missing value.
Fractional exponents work as roots — 81/3 is the cube root of 8. Negative exponents give reciprocals — 2−3 is 1/8.
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How exponents work
An exponent tells you how many times to multiply a base by itself: 25 means 2 × 2 × 2 × 2 × 2 = 32. The base is the number being multiplied, the exponent (or power, or index) is the count. That definition is simple enough while the exponent is a positive whole number — the useful part is what happens when it is not.
Negative exponents
A negative exponent means the reciprocal of the positive version: b−n = 1 ÷ bn. So 2−3 = 1/23 = 1/8 = 0.125. Negative exponents never produce negative results — that is the single most common mistake with them. 10−6 is 0.000001, a small positive number.
Fractional exponents are roots
b1/n is the n-th root of b. So 91/2 = √9 = 3, and 81/3 = ∛8 = 2. A fraction with a numerator other than 1 does both jobs: bm/n = (ⁿ√b)m. For example 82/3 = (∛8)² = 2² = 4. Note that a fractional exponent on a negative base is usually undefined in the real numbers — (−8)1/2 has no real answer, though (−8)1/3 = −2 does.
Zero and one
Any non-zero number to the power 0 equals 1 — including 0.50 and (−7)0. Any number to the power 1 is itself. The case 00 is left undefined here, since context decides whether it should be 1 or indeterminate.
Laws of exponents
These are the rules that let you simplify expressions without computing them: bm × bn = bm+n (add when multiplying the same base), bm ÷ bn = bm−n (subtract when dividing), (bm)n = bmn (multiply for a power of a power), (ab)n = anbn, and (a/b)n = an/bn.
Solving for the exponent
If you know the base and the result but need the exponent, that is a logarithm: n = log(result) ÷ log(base). This calculator does it for you in the "find the exponent" mode. It is the same maths behind questions like "how many years until my investment doubles" — see the compound interest calculator for that applied version.
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Frequently asked questions
What does a negative exponent mean?
It means take the reciprocal: b to the power −n equals 1 divided by b to the power n. So 5⁻² = 1/25 = 0.04. The result stays positive — a negative exponent shrinks a number, it does not make it negative.
How do I calculate a fractional exponent?
A fractional exponent is a root. b^(1/n) is the n-th root of b, and b^(m/n) is the n-th root of b raised to the power m. For example 16^(3/4) = (⁴√16)³ = 2³ = 8. Enter the fraction as a decimal (3/4 becomes 0.75) or use the root mode.
Why is anything to the power of zero equal to 1?
Because of the division law: b^n ÷ b^n = b^(n−n) = b⁰, and any non-zero number divided by itself is 1. So b⁰ must equal 1. The one exception is 0⁰, which is left undefined.
Can I raise a negative number to a power?
Yes for whole-number exponents — a negative base gives a positive result with an even exponent and a negative result with an odd one, so (−3)² = 9 but (−3)³ = −27. Fractional exponents on negative bases have no real answer unless the denominator of the fraction is odd.
How do I find the exponent if I know the answer?
Use a logarithm: exponent = log(result) ÷ log(base). Switch this calculator to 'Exponent' mode and it does that for you. For example with base 2 and result 1024, the exponent is log(1024)/log(2) = 10.
What is scientific notation and when does it appear?
Scientific notation writes a number as a value between 1 and 10 multiplied by a power of ten — 4.5 × 10⁸ instead of 450,000,000. This calculator shows it whenever a result is very large or very small, because the plain decimal form becomes unreadable.