Exponent Calculator
Raise a number to any power, including the negative exponents that give reciprocals and the fractional ones that give roots.
Updated
Negative gives a reciprocal; a fraction such as 0.5 gives a root.
Result
1,024
2 to the power of 10
- Reads as
- 2^10
- Reciprocal
- no
- A root
- no
In short
What does a negative exponent mean?
A reciprocal, not a negative answer. 2^-3 is 1 divided by 2^3, which is 0.125 — never −8. A fractional exponent is a root instead, so 9^0.5 is 3. Only the ordinary case, a positive whole number, is repeated multiplication.
The minus sign moves the number under a fraction bar. It never changes its sign.
How to use the exponent calculator
Type a base and an exponent. The result appears with a line describing which rule produced it, because the arithmetic is trivial and knowing which of three quite different operations just happened is not.
A positive whole exponent is repeated multiplication and nobody struggles with it: 2^10 is ten twos multiplied together, which is 1,024. The other two cases look like the same notation and do something else entirely, and the notation gives no hint.
1,024
2 to the power of 10
repeated multiplication
0.125
2 to the power of −3
a reciprocal, not −8
3
9 to the power of 0.5
a square root
A negative exponent means one over the positive power. 2^-3 is 1/2^3, which is one eighth. The result of a negative exponent on a positive base is always positive and always smaller than one — the minus sign moves the number under a fraction bar rather than across zero.
A fractional exponent is a root. The denominator says which root: x^(1/2) is the square root, x^(1/3) is the cube root, x^(2/3) is the cube root squared. That is why 9^0.5 is 3 rather than 4.5, which is the answer people expect when they read the exponent as a multiplier.
- 2^0
- 1 — anything to the zero
- 2^1
- 2
- 2^3
- 8
- 2^10
- 1,024
- 2^-1
- 0.5
- 2^-3
- 0.125
- 2^0.5
- about 1.414 — the square root
Read down the list and the pattern is one operation, not three: each step down the exponent divides by the base. Two, one, a half, a quarter. The negative exponents are simply the same ladder continued past zero, which is also why anything to the zero is one.
The root written the other way
The square root page takes the same operation in its familiar notation, with the simplified radical form a fractional exponent cannot show.
Open square root calculator →The three kinds of exponent side by side on familiar bases, with the operation each one actually performs rather than just its result.
| Expression | Result | What it does |
|---|---|---|
| 10^6 | 1,000,000 | repeated multiplication |
| 5^3 | 125 | repeated multiplication |
| 2^10 | 1,024 | repeated multiplication |
| 2^0 | 1 | the empty product |
| 10^-2 | 0.01 | reciprocal of 10^2 |
| 2^-3 | 0.125 | reciprocal of 2^3 |
| 9^0.5 | 3 | square root |
| 27^(1/3) | 3 | cube root |
| (−8)^(1/3) | −2 | cube root, odd so it exists |
| (−9)^0.5 | not real | even root of a negative |
| 1.05^10 | 1.6288946 | compound growth over 10 periods |
Why anything to the zero is one
It looks like a convention invented to avoid a gap, and it is not. It falls out of the rule that dividing powers subtracts exponents: x^3 divided by x^3 is x^0, and any non-zero number divided by itself is one. So x^0 has to be one for the rest of the arithmetic to hold together.
- 2^3
- 8
- 2^2
- 4 — divided by 2
- 2^1
- 2 — divided by 2
- 2^0
- 1 — divided by 2
- 2^-1
- 0.5 — divided by 2
- One consistent rule
- each step divides
Nothing special happens at zero, and nothing special happens crossing into the negatives. Each row is the one above divided by the base, which produces both the zero rule and the reciprocal rule without either being invented separately.
Zero to the zero is the one genuine exception, and it is left undefined rather than settled. Approached one way it looks like one; approached another it looks like zero, and different fields adopt different conventions for their own convenience.
When a root of a negative number exists
An odd root of a negative number is perfectly ordinary: the cube root of −8 is −2, because −2 multiplied by itself three times gives −8. An even root is not, because multiplying any real number by itself an even number of times gives a positive result.
Do
- Read a negative exponent as one over the positive power
- Read a fractional exponent as a root, with the denominator naming it
- Expect an odd root of a negative number to be negative
- Use brackets when the base is negative, since −2^2 and (−2)^2 differ
Don't
- Expect a negative exponent to produce a negative answer
- Read 9^0.5 as nine times a half
- Look for a real square root of a negative number
- Assume a calculator handles the cube root of a negative base
That last point is a real limitation rather than a quibble. Most programming languages return an error for the cube root written as a fractional power, because a float cannot distinguish an exact third from 0.3333. This tool detects the intended fraction and answers −2, which is why it can do something a plain power function cannot.
The formula, worked line by line
Three rules that are really one rule seen from different sides. Each step up an exponent multiplies by the base, and each step down divides by it — everything else follows.
“A negative exponent moves the number under a fraction bar. It does not move it across zero.”
Almost every mistake on this subject is that sentence not being known, so it is worth reading twice rather than the formulas below.
positive whole: x^n = x multiplied by itself n times
zero: x^0 = 1 for any non-zero x
negative: x^-n = 1 / x^n
fractional: x^(1/n) = the nth root of x
2^-3 = 1 / 2^3 = 1 / 8 = 0.125- Five percent a year
- multiply by 1.05
- Over ten years
- 1.05^10
- Growth factor
- 1.6288946
- On 10,000
- 16,288.95
- Simple 5 percent x 10 would give
- 15,000
- The exponent is worth
- 1,288.95 more
The gap between multiplying by ten and raising to the tenth is what compounding is, and it is why every interest, inflation and growth calculation on this site runs through a power rather than a multiplication.
Why brackets matter on a negative base
Because the exponent binds tighter than the minus sign. Written without brackets, −2^2 means the negative of 2^2, which is −4. Written as (−2)^2 it means the square of −2, which is 4. Both are correct readings of different expressions, and the brackets are the only thing distinguishing them.
The three cases, and what decides the sign
- Positive base, any exponent
- always positive
- Negative base, even whole exponent
- positive
- Negative base, odd whole exponent
- negative
- Negative base, even root
- no real answer
- A negative EXPONENT
- never affects the sign
The last row is the one to keep. The sign of the answer is decided by the base and by the parity of a whole exponent. A minus in the exponent controls the size, moving the result below one, and has no say in the sign at all.
One habit: when an exponent is negative, say "one over" out loud before doing anything. It converts the hardest case into the easiest one and removes the error entirely.
Questions people ask
What does a negative exponent mean?
One over the positive power. 2^-3 means 1 divided by 2^3, which is one eighth or 0.125. It does not mean −8. A negative exponent makes the result smaller than one for a base above one, but it has no effect whatsoever on the sign of the answer.
What is 2 to the power of −3?
0.125, which is one eighth. Work out 2^3 first, giving 8, then take one over it. The most common error is answering −8, which comes from reading the minus sign as belonging to the result rather than as an instruction to take a reciprocal.
Why is anything to the power of zero equal to one?
Because dividing powers subtracts exponents, so x^3 divided by x^3 is x^0, and anything non-zero divided by itself is one. It also falls out of the ladder: each step down an exponent divides by the base, so 2^1 is 2 and 2^0 is 2 divided by 2.
What does a fractional exponent do?
It takes a root, with the denominator naming which one. x^(1/2) is the square root, x^(1/3) is the cube root, and x^(2/3) is the cube root squared. So 9^0.5 is 3, not 4.5 — the exponent is not a multiplier and never behaves like one.
Can you take the cube root of a negative number?
Yes. The cube root of −8 is −2, because multiplying −2 by itself three times gives −8. Odd roots of negative numbers are ordinary; even roots are not, because no real number multiplied by itself an even number of times produces a negative result.
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