The straightforward case: 2⁵
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2^5Five twos multiplied: 2 × 2 × 2 × 2 × 2 = 32. This is the only case where the exponent really is a count of factors, and it is worth writing out once so that the cases below can be seen as departures from it.
Exponent Calculator
Powers with the rule named, including the negative and fractional ones.
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Want to solve these without the calculator?
The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
An exponent counts factors - until it stops being a count
For a whole exponent there is nothing to explain: 2⁵ means five twos multiplied together, which is 32. The interesting cases are the ones where the exponent is no longer a count of factors, and there are three of them: zero, negative, and fractional. Each has one rule behind it, and each rule follows from the same fact - that multiplying powers of the same base adds their exponents.
Zero: 2³ ÷ 2³ is 1, and subtracting the exponents gives 2⁰, so 2⁰ must be 1 - for every base except 0 itself. Negative: 2³ ÷ 2⁵ is 1/4, and subtracting gives 2⁻², so a negative exponent is a RECIPROCAL and never a negative answer. Fractional: 2^(1/2) times 2^(1/2) is 2¹, so 2^(1/2) is whatever squares to 2 - the square root.
Everything here is exact. 2⁻³ is 1/8 rather than 0.125, (2/3)³ is 8/27, 9^(1/2) is exactly 3, and 8^(1/2) stays 2√2 instead of collapsing to 2.828. A decimal is offered beside the exact answer whenever one exists, and labelled as an approximation whenever it is one.
What to notice
- A negative exponent gives a reciprocal, not a negative number. 2⁻³ is 1/8 and is positive; only a negative BASE can make an answer negative.
- Zero as an exponent gives 1 for every base except zero. 0⁰ has no agreed value, and 0 to a negative power would divide by zero.
- A fractional exponent is a root: the denominator says which root and the numerator is the power. 8^(2/3) is the cube root of 8, squared, which is 4.
- A fraction raised to a power raises both parts. (2/3)³ is 8/27 - the denominator is cubed too, which is the step most often skipped.
- The sign of a negative base is decided by the exponent alone: an even exponent pairs the minus signs off and gives a positive answer, an odd one leaves one over.
- Brackets change everything. (−2)⁴ is 16 but −2⁴ is −16, because without brackets the power binds tighter than the minus sign.
How to work out a power
Check the exponent's sign first
If it is negative, write the reciprocal and make the exponent positive: 2⁻³ becomes 1/2³. Doing this first stops the classic mistake of answering −8.
Check whether it is a fraction
A fractional exponent is a root. The bottom of the fraction chooses the root and the top is the power: x^(m/n) is the n-th root of x^m.
Deal with a fractional base
Raise the numerator and the denominator separately. (2/3)³ is 2³ over 3³, which is 8/27.
Multiply out what is left
A whole exponent is now just repeated multiplication. Keep the sign in mind: an odd number of negative factors leaves a negative answer.
Finish the reciprocal, if there was one
Flip the result you just computed. 1/2³ is 1/8 - and check that the answer is smaller than 1, which is what a negative exponent on a base bigger than 1 always gives.
Powers and the rule behind each
The four cases that are not simply repeated multiplication, alongside the ones that are.
| Power | Rule | Exact answer | As a decimal |
|---|---|---|---|
| 2⁵ | repeated multiplication | 32 | 32 |
| 2⁰ | anything except 0 to the power 0 is 1 | 1 | 1 |
| 2⁻³ | negative exponent is a reciprocal | 1/8 | 0.125 |
| (2/3)³ | raise top and bottom | 8/27 | 0.296… |
| 9^(1/2) | fractional exponent is a root | 3 | 3 |
| 8^(1/2) | root simplified, not rounded | 2√2 | 2.828… |
| (−2)⁴ | even exponent, minus signs pair off | 16 | 16 |
| (−2)³ | odd exponent, one minus left over | −8 | −8 |
Worked examples
A negative exponent: 2⁻³
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2^-3Not −8. A negative exponent means a reciprocal: 2⁻³ is 1/2³ = 1/8. The reason is division - 2² ÷ 2⁵ is 1/8, and subtracting the exponents gives 2⁻³, so the two must be equal. The answer is positive, and smaller than 1.
A fraction as the base: (2/3)³
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(2/3)^3Both parts are cubed: 2³ over 3³, which is 8/27. Cubing only the numerator - a very common slip - would give 8/3, which is more than 8 times too big. As a decimal it is 0.296… and never terminates, so the fraction is the honest answer.
A fractional exponent: 9^(1/2)
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9^1/2A power of 1/2 is a square root, so this is √9 = 3 exactly. The reason: 9^(1/2) times 9^(1/2) adds the exponents to give 9¹, so 9^(1/2) must be the number that squares to 9.
When the root does not come out: 8^(1/2)
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8^1/2√8 is irrational, so the answer stays a radical - simplified to 2√2 by taking the square factor 4 outside. The decimal 2.828… sits beside it as an approximation, never in place of it.
The sign of a negative base: (−2)⁴ and (−2)³
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(-2)^4Four negative factors pair up into two positive products, so (−2)⁴ = 16. Three leave one minus sign unpaired, so (−2)³ = −8. The exponent alone decides the sign - and note that −2⁴ without brackets means −(2⁴) = −16, which is a different question.
Common mistakes
- Reading a negative exponent as a negative answer. 2⁻³ is 1/8, not −8: the minus sign moves the power to the denominator, it does not travel to the answer.
- Multiplying the base by the exponent. 2⁵ is 32, not 10. The exponent counts factors; it is not a multiplier.
- Raising only the numerator of a fraction. (2/3)³ is 8/27, not 8/3 - the denominator is cubed too.
- Dropping the brackets round a negative base. (−2)⁴ is 16 while −2⁴ is −16, because a power binds tighter than a minus sign.
- Thinking 0⁰ is 1 or 0. It has no agreed value; different fields choose different conventions and this page refuses rather than picking one.
- Rounding a root instead of simplifying it. 8^(1/2) is 2√2 exactly; 2.83 is an approximation, and it stops being useful the moment the value is squared again.
Exponent FAQ
What does a negative exponent mean?
A reciprocal. x⁻ⁿ is 1/xⁿ, so 2⁻³ is 1/8 - a positive number smaller than 1. It follows from dividing powers: 2² ÷ 2⁵ is 1/8 and subtracting exponents gives 2⁻³, so the two have to be the same thing.
Why is anything to the power zero equal to 1?
Because dividing a power by itself gives 1, and subtracting the exponents gives zero: 2³ ÷ 2³ = 1 and also = 2⁰. The one exception is 0⁰, which has no agreed value - the pattern that forces 1 needs a base you are allowed to divide by.
What does a fractional exponent mean?
A root. The denominator says which root and the numerator is a power: x^(1/2) is the square root, x^(1/3) the cube root, and x^(2/3) is the cube root of x squared. It follows from multiplying powers: x^(1/2) times x^(1/2) is x¹.
How do I raise a fraction to a power?
Raise the numerator and the denominator separately: (2/3)³ = 2³/3³ = 8/27. Doing it to only the top is the most common error here, and it is out by a factor of the denominator to that power.
Is (−2)⁴ the same as −2⁴?
No, and this is worth memorising. (−2)⁴ raises −2 to the fourth and gives +16. Without brackets, −2⁴ means −(2⁴) = −16, because exponentiation happens before the minus sign is applied. Bracket a negative base every time.
When is a power of a negative number positive?
When the exponent is even. Each pair of negative factors multiplies to a positive, so an even count leaves nothing negative behind and an odd count leaves exactly one minus sign. (−3)² is 9 and (−3)³ is −27.
What is 2 to the power 1/2?
The square root of 2, which is irrational - so the exact answer is √2 and 1.414… is only an approximation of it. This page keeps the radical and simplifies it where it can: 8^(1/2) is written 2√2 rather than left as √8.
Can this calculator do cube roots?
Yes. A fractional exponent is taken exactly as a root of any order up to the 12th: 8^(1/3) is 2, 16^(3/4) is 8, and 54^(1/3) stays 3∛2 rather than becoming a decimal. An odd root of a negative base works too ((-8)^(1/3) is -2); an even root of a negative number is not a real number, and the page says so.
