Exponent Calculator
Powers, roots, and the laws that connect them - including the cases that trip people up, like negative bases and fractional exponents.
Last updated: August 2026 · Written and verified by Akash Pandey
Calculate a power or root
Result
This runs the same calculation engine as the main scientific calculator, entirely in your browser. Nothing you type is sent anywhere.
What an exponent means
An exponent is repeated multiplication. 2^5 means five 2s multiplied together: 2×2×2×2×2 = 32. The base is the number being multiplied, and the exponent (or index, or power) counts how many times.
bⁿ = b × b × … × b (n copies of b)
- b is the base
- n is the exponent
- When n is not a positive whole number, the definition is extended by the index laws below.
The index laws
| Law | Rule | Example |
|---|---|---|
| Product | bᵐ × bⁿ = bᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 2⁵ ÷ 2² = 2³ = 8 |
| Power of a power | (bᵐ)ⁿ = bᵐⁿ | (2³)² = 2⁶ = 64 |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2×3)² = 4×9 = 36 |
| Zero exponent | b⁰ = 1 (b ≠ 0) | 7⁰ = 1 |
| Negative exponent | b⁻ⁿ = 1 ÷ bⁿ | 2⁻³ = 1/8 = 0.125 |
| Fractional exponent | b^(1/n) = ⁿ√b | 8^(1/3) = 2 |
| General fractional | b^(m/n) = ⁿ√(bᵐ) | 8^(2/3) = 4 |
The zero and negative rules are not arbitrary conventions - they fall out of the quotient law. 2³ ÷ 2³ is obviously 1, and by the rule it is 2⁰, so 2⁰ must be 1. Likewise 2³ ÷ 2⁵ = 1/4, and the rule gives 2⁻², so 2⁻² must be 1/4.
Worked examples
Evaluate 2^10
2^10
- This is ten 2s multiplied together.
- Doubling repeatedly: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.
- Worth memorising - 2^10 = 1024 is why a kilobyte is 1024 bytes.
= 1024
Evaluate 8^(2/3)
8^(2/3)
- Split the fractional exponent: the denominator is a root, the numerator a power.
- Take the cube root first:
∛8 = 2. - Then square it:
2² = 4. - Rooting first keeps the numbers small; squaring first gives
∛64, which is the same answer by a harder path.
= 4
Evaluate 5^(−2)
5^(-2)
- A negative exponent means the reciprocal:
5⁻² = 1 ÷ 5². 5² = 25.1 ÷ 25 = 0.04.
= 0.04
- A negative exponent means the reciprocal:
Solve 2ⁿ = 1024
log(1024, 2)
- An unknown in the exponent calls for a logarithm.
- Take the base-2 logarithm of both sides:
n = log₂(1024). - Since 1024 is 2 multiplied by itself 10 times, the answer is 10.
= 10
The cases that catch people out
Negative bases need brackets
−5² and (−5)² are different. The first squares 5 and then negates it, giving −25, because exponentiation binds more tightly than the unary minus. The second squares −5, giving 25. Every calculator that follows standard precedence behaves this way, including this one - see the order of operations guide.
Fractional powers of negative numbers are usually undefined
The square root of a negative number has no real value, so (−8)^(1/2) is undefined here. (−8)^(1/3) is a subtler case: mathematically the real cube root is −2, but computing it as a general power runs through a complex intermediate. Use the dedicated cube-root function ∛ on the main calculator when you want the real root of a negative number.
0⁰ is a special case
The limit arguments point in two directions at once: x⁰ tends to 1 while 0ˣ tends to 0. In practice nearly all computing environments define 0⁰ = 1, because that is what makes power series and combinatorial formulas work, and this calculator follows that convention.
Powers of 2 and 10 for reference
| n | 2ⁿ | 10ⁿ |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 2 | 10 |
| 2 | 4 | 100 |
| 3 | 8 | 1,000 |
| 4 | 16 | 10,000 |
| 5 | 32 | 100,000 |
| 6 | 64 | 1,000,000 |
| 8 | 256 | 100,000,000 |
| 10 | 1,024 | 10,000,000,000 |
| 16 | 65,536 | 10¹⁶ |
| 20 | 1,048,576 | 10²⁰ |
| 32 | 4,294,967,296 | 10³² |
Frequently Asked Questions
What does a negative exponent mean?
It means the reciprocal of the positive power: b⁻ⁿ = 1 ÷ bⁿ. So 2⁻³ = 1/8 = 0.125. A negative exponent never makes the result negative.
Why is any number to the power 0 equal to 1?
It follows from the quotient law. bⁿ ÷ bⁿ is 1, and the law says it equals b⁰, so b⁰ must be 1. The one exception is 0⁰, which is a special case defined as 1 by convention in most computing contexts.
What does a fractional exponent do?
The denominator takes a root and the numerator applies a power: b^(m/n) = ⁿ√(bᵐ). So 8^(2/3) is the cube root of 8, squared, which is 4.
Why does −5² give −25 instead of 25?
Exponentiation binds more tightly than the minus sign, so the expression reads as "the negative of 5 squared". Write (−5)² if you want the negative number itself to be squared.
How do I solve for an exponent?
Use a logarithm. If bⁿ = x, then n = log(x) ÷ log(b), which the "Solve bⁿ = x" mode above does for you. See the log calculator for more.
What is the difference between a power and a root?
They are inverse operations. 2⁵ = 32 says five 2s multiply to 32; ⁵√32 = 2 asks which number, multiplied by itself five times, gives 32. A root is just a fractional power.
What happens when the base is negative and the exponent is fractional?
A fractional exponent with an even root of a negative number is not a real number. For example, (-8)^(1/3) works because the cube root of -8 is -2, but (-8)^(1/2) gives an imaginary result because you cannot take the square root of a negative in the real number system.
How do I calculate very large or very small powers?
Use logarithms. For massive numbers like 100^50, compute 50 × log(100) and convert back. For tiny results like 0.5^100, use 100 × log(0.5). The calculator above handles this automatically.