Exponent Calculator
Raise any base to any power — negative bases, negative exponents, and roots written as fractions. You get the answer in plain digits and in scientific notation, plus the two inverses: the exponent recovered by logarithm and the base recovered by root.
Base and exponent
The number being raised. Negative bases are allowed, and are where fractional exponents get interesting.
How many times the base is used. Write a root as a fraction — 1/3 for a cube root — so the exact value is what gets used.
10 copies of 2 multiplied together.
- Result
- 1,024
- Scientific notation
- 1.024 × 10³
- Reciprocal (the same base to −b)
- 0.0009765625
- Exponent recovered by logarithm
- 10
- Base recovered by root
- 2
Three rules, and everything else derived from them
An exponent starts as shorthand for repeated multiplication: 2¹⁰ is ten 2s multiplied together. Three rules follow from that reading alone, and once you have them, the awkward cases — zero exponents, negative ones, fractional ones — stop being separate facts to memorise and become consequences.
aᵐ · aⁿ = aᵐ⁺ⁿ • aᵐ ÷ aⁿ = aᵐ⁻ⁿ • (aᵐ)ⁿ = aᵐⁿ- Product
- m copies followed by n copies is m + n copies in one product
- Quotient
- Each copy underneath cancels one copy above it
- Power of a power
- n copies of a block of m copies is m × n copies
Zero comes first. By the quotient rule aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and anything nonzero divided by itself is 1. So a⁰ = 1 — not because someone decided it, but because the rule leaves no other option.
The negative exponent falls out of the same rule one step later. Apply it to a⁰ ÷ aⁿ and the answer is a⁰⁻ⁿ = a⁻ⁿ. Now evaluate the left side directly: a⁰ is 1, so it is 1 ÷ aⁿ. The same expression has two names, therefore a⁻ⁿ = 1 ÷ aⁿ. A negative exponent is a reciprocal, and that is a derivation rather than a rule to remember.
Fractions come from the power-of-a-power rule. Whatever a^(1/n) is, raising it to the n gives a^(n/n) = a. The number that gives back a when raised to the n is the nth root, so a^(1/n) is exactly that root. A general fraction a^(p/q) is then the qth root taken p times, and the order does not matter.
That last rule is where a negative base becomes delicate. The qth root of a negative number is real only when q is odd: (−2)³ = −8, so the cube root of −8 is −2, but nothing squared gives a negative number, so the square root of −8 is not a real number at all. The sign of the answer then depends on p — an odd numerator keeps the minus, an even one squares it away, which is why (−8)^(1/3) is −2 while (−8)^(2/3) is 4. This calculator recovers the exact fraction behind the exponent to decide which case applies, and refuses the ones that leave the real numbers rather than returning NaN.
2¹⁰, and then the awkward relatives
Ten 2s multiplied together is 1024. In scientific notation that is 1.024 × 10³. The two inverses check it: log 1024 ÷ log 2 = 10 recovers the exponent, and the tenth root of 1024 is 2, which recovers the base. Those are the only two questions you can ask backwards from a power, and a logarithm is nothing more than the first of them.
Flip the exponent to −10 and the answer is not negative, it is small: 2⁻¹⁰ = 1 ÷ 1024 = 0.0009765625, or 9.765625 × 10⁻⁴. Multiply the two results together and you get 1, which is the reciprocal relationship stated as arithmetic.
Now the case that separates calculators. Enter a base of −8 and an exponent of 1/3. The exact fraction has denominator 3, which is odd, so a real cube root exists: 8^(1/3) = 2, and the odd numerator keeps the sign, giving −2. Check it — (−2) × (−2) × (−2) = −8. Change the exponent to 2/3 and the even numerator drops the sign for 4. Change it to 1/2 and there is no real answer to give, so the calculator says so and explains which rule ran out, rather than printing NaN. Type 0.333333 in place of 1/3 and you get the same refusal, correctly: that number is 333333/1000000, its denominator is even, and it is not one third.
What this calculator assumes
- 0⁰ is 1. That is the combinatorial and algebraic convention, and what IEEE 754 specifies. Treat it as indeterminate if you are evaluating a limit.
- Answers stay in the real numbers. A negative base with an even root, or with an irrational exponent, is refused with an explanation rather than answered in the complex plane.
- A fractional exponent is recognised as a fraction only if you write it as one, or if the decimal you type is exactly a fraction with a denominator up to a million. 1/3 works; 0.333333 is a different number and is treated as one.
- Results are kept to twelve significant figures, which is inside what the underlying arithmetic can carry honestly. Whole numbers below a trillion are returned exactly.
- A result too large or too close to zero to represent is refused rather than reported as infinity or as nought. The usable range is roughly 10⁻³⁰⁷ to 10³⁰⁸.
- The base field holds the entire base. Entering −2 with an exponent of 4 means (−2)⁴ = 16, not −2⁴ = −16.
Exponent FAQ
Is 0 to the power 0 equal to 1?
This calculator says 1, and that is a convention rather than a discovered fact. Approach the point from one direction and x⁰ is 1 for every x on the way, so the limit is 1. Approach it from the other and 0ˣ is 0 for every positive x, so the limit is 0. Analysis therefore calls 0⁰ an indeterminate form and refuses to assign it a value. Algebra and combinatorics cannot afford that: 0⁰ = 1 is what makes the empty product work, what makes the binomial theorem correct at x = 0, and what lets a polynomial be written as a sum of terms aₙxⁿ without carving out an exception. IEEE 754-2019, the arithmetic standard your computer follows, specifies pow(±0, ±0) = 1 for the same practical reasons. If you are working with limits rather than with algebra, treat the form as indeterminate and do the limit properly.
Why does (−8)^(1/3) come back as an error on most calculators?
Because the exponent arrives as a decimal with no memory of having been a fraction. One third stored in binary is 0.333333333333333314829616256247, and no real number raised to that power gives a negative answer — the rule is defined only for exponents that reduce to a fraction with an odd denominator, and the machine cannot see a denominator at all. So the standard pow function returns NaN and the calculator prints an error, even though the cube root of −8 is the perfectly ordinary real number −2. This tool recovers the exact fraction the exponent stands for using continued fractions, checks whether the denominator in lowest terms is odd, and if it is, takes the root of the magnitude and puts the sign back. That is also why the exponent field accepts 1/3 as text: type 0.333333 instead and you get an honest refusal, because 0.333333 really is a different number with no real answer for a negative base.
What does a negative exponent mean?
A reciprocal, not a negative answer. 2⁻¹⁰ is 1 ÷ 2¹⁰ = 0.0009765625, a small positive number. It is not something to memorise — it follows from the quotient rule. Dividing powers subtracts exponents, so a⁰ ÷ aⁿ is a⁰⁻ⁿ = a⁻ⁿ. But a⁰ is 1, so the same expression is 1 ÷ aⁿ. Two names for one quantity, therefore a⁻ⁿ = 1 ÷ aⁿ. The sign of the base is untouched by any of this: (−2)⁻³ is 1 ÷ (−8) = −0.125.
What is the difference between (−2)⁴ and −2⁴?
The parentheses, and the answers differ by a sign. (−2)⁴ raises the negative number to the fourth power: (−2)(−2)(−2)(−2) = 16. Written without parentheses, −2⁴ means the negative of 2⁴, because exponentiation binds tighter than the minus sign, so it is −(16) = −16. This calculator takes the base field as the whole base, parentheses included, so entering −2 with an exponent of 4 gives 16. If you want −16, negate the result yourself or enter it as the answer to 2⁴ with a minus in front.
How do I read a result like 1.024 × 10³?
Move the decimal point as many places as the exponent says: three places right gives 1024. A negative exponent moves it left, so 9.765625 × 10⁻⁴ is 0.0009765625. Scientific notation is shown here for every result and used in place of plain digits once a number passes about a thousand trillion or drops below a millionth, because past that point the digits are mostly zeros and the twelve significant figures this calculator keeps are all the information there is. The coefficient is always at least 1 and less than 10, which is what makes two numbers in this form easy to compare at a glance.
Sources and review notes
- NIST Digital Library of Mathematical Functions §4.2 — the definition of a general power and the branch conditions that decide when one is real
- IEEE Std 754-2019, Standard for Floating-Point Arithmetic — specifies pow(±0, ±0) = 1 and the behaviour of pow on a negative base
- OpenStax, College Algebra 2e §1.2 — the product, quotient and power rules, and scientific notation
The rules on this page are theorems, not measurements, so there is nothing here to go out of date. The sources are cited for the two places where a choice was made rather than derived: the value given to 0⁰, and the decision to answer only in the real numbers.