A negative exponent means the reciprocal, not a negative answer: 2⁻³ is 1/8. A fractional exponent means a root, with the denominator naming which one: 8^(2/3) is the cube root of 8, squared, which is 4. And anything to the power zero is 1, because aⁿ ÷ aⁿ is both 1 and a⁰ — the only case in dispute is 0⁰.
Negative exponents
2⁻³ is one eighth. Not minus eight, and not minus one eighth. The minus sign flips the fraction over; it has nothing to do with the sign of the answer.
It follows from subtracting exponents. a³ ÷ a⁵ is a⁻² by the rule, and it is also 1/a² by cancelling. Both readings must agree, which forces the definition rather than choosing it.
The sign of the answer comes from the base alone. (−2)⁻³ is −1/8, negative because the base is negative and the exponent is odd — not because of the minus in the exponent.
Fractional exponents
The denominator is the root and the numerator is the power. 8^(2/3) is the cube root of 8, then squared: 2² = 4.
Order matters for the arithmetic, not the answer. Squaring first gives the cube root of 64, which is also 4 — but you had to find the cube root of a much bigger number. Take the root first and the numbers stay small.
This is not a different operation bolted on. It is the only definition that keeps (a^(1/2))² = a¹ working, which is exactly why roots and powers are the same notation.
Why anything to the zero is one
aⁿ ÷ aⁿ is obviously 1. By the subtraction rule it is also a⁰. So a⁰ has to be 1, for every a that is not zero.
The pattern says the same thing going down: 2⁴ = 16, 2³ = 8, 2² = 4, 2¹ = 2, each half the last. The next step down is 1.
It is not a convention adopted for tidiness. Any other value would break the division rule at its first application, and every polynomial with a constant term would need a special case.
0⁰, and why nobody agrees
Two good rules collide. Anything to the power zero is 1. Zero to any power is 0. They disagree only at 0⁰, and both have a claim.
Most of mathematics defines 0⁰ = 1, because that is what makes the binomial theorem, power series and the empty product work without exceptions. x⁰ is 1 everywhere else, and the empty product convention says a product of no factors is 1.
In analysis it is left undefined, because x^y has no single limit as both approach zero — approach along different paths and you get different answers, so no continuous extension exists.
This page reports both rather than picking one. It is the honest answer, and a tool that silently returns 1 has hidden a real disagreement.
The bracket trap
−7⁰ is −1, and (−7)⁰ is 1. The difference is entirely the brackets, and it is a notation rule rather than a mathematical one.
Exponentiation binds tighter than the minus sign, so −7⁰ reads as −(7⁰) = −1. Only the brackets put the minus inside the base.
The same trap catches −2⁴, which is −16 rather than 16. Spreadsheets disagree with mathematics here — Excel evaluates −2^4 as 16 — which is worth knowing before trusting a formula that mixes the two.
Where a double gives up
A JavaScript number holds about seventeen significant digits. 2²⁰⁰ has sixty-one, so a floating-point answer shows a handful of them and an exponent, and everything after the seventeenth is invented.
This page computes whole-number powers on integers of unlimited size. Every digit shown is real, and the page says how many there are so the difference is visible.
It matters wherever exactness does — cryptography, combinatorics, or simply checking a homework answer that a calculator rendered in scientific notation and thereby made unverifiable.
Sources and methodology
The laws are elementary; the 0⁰ question is genuinely open and worth sourcing properly.
Method. Whole-number powers are computed on BigInt by repeated multiplication, so the answer is exact at any size rather than a double with an exponent. Negative exponents are reported as exact fractions; fractional ones take the root first, on integers, and are given as a surd where one exists. 0⁰ is not computed at all — both defensible positions are reported. That engine is verified on every change against 68 hand-written assertions, including that every power from base −12 to 12 and exponent 0 to 30 matches an independently written BigInt loop, and that aᵐ × aⁿ = aᵐ⁺ⁿ holds on every computed value. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The inverse question — what power gives this number — is the Log Calculator, where every law here reappears in its logarithmic form. For roots specifically, the Square Root and Cube Root pages give the exact surd.