Exponent Calculator
Raise any base to any power — including negative, zero, and fractional exponents — and see the result with the rule that explains what the exponent means.
How to use this calculator
- 1Enter the base — the number being multiplied.
- 2Enter the exponent — how many times, or which root, to apply.
- 3Read the result, shown in scientific notation when it grows very large.
- 4Check the plain-language meaning to confirm you set it up correctly.
How it works
Exponentiation
bⁿ = b × b × … × b (n times, for whole n) b⁰ = 1, b⁻ⁿ = 1 ÷ bⁿ b^(1/n) = the nth root of b b^(m/n) = the nth root of b, raised to m
Exponentiation is repeated multiplication generalised. For a whole-number power, bⁿ simply means multiplying the base b by itself n times. From that single idea the other cases follow by keeping the rules consistent: b⁰ must equal 1 so that the pattern bⁿ ÷ b = bⁿ⁻¹ keeps working down to zero; a negative exponent flips to a reciprocal, b⁻ⁿ = 1 ÷ bⁿ, for the same reason; and a fractional exponent is a root, because b^(1/2) squared has to give b, which is exactly what the square root does. These extensions are not arbitrary — they are the only definitions that let the laws of exponents hold everywhere.
Worked example
2 raised to the 10th power is 2 multiplied by itself ten times: 2 × 2 × … = 1,024. By the same rules, 2⁻² is the reciprocal 1 ÷ 4 = 0.25, and 9^(1/2) is the square root of 9, which is 3.
Exponent Calculator: the complete guide
Reading exponents beyond whole numbers
A whole-number exponent is easy to picture: 3⁴ is 3 × 3 × 3 × 3 = 81, four copies multiplied together. The confusion begins with the cases that don't fit that picture — what could it mean to multiply a number by itself zero times, or a negative number of times, or half a time? The answer is that mathematicians chose definitions for these cases specifically so that the familiar rules keep working, which turns out to give each one a concrete meaning.
A zero exponent always yields 1 (for any non-zero base), because dividing bⁿ by b repeatedly steps the exponent down, and reaching zero lands on 1. A negative exponent is a reciprocal: b⁻² is 1 ÷ b². And a fractional exponent is a root: b^(1/3) is the cube root of b, since cubing it must return b. Once you see these as the consistent continuation of the whole-number pattern rather than separate rules to memorise, exponents stop being a collection of special cases.
The laws that make exponents powerful
The reason exponents appear everywhere in mathematics and science is a small set of laws that make them easy to manipulate. Multiplying powers of the same base adds the exponents (bᵐ × bⁿ = bᵐ⁺ⁿ); dividing subtracts them; raising a power to a power multiplies them ((bᵐ)ⁿ = bᵐⁿ). These turn otherwise fearsome multiplications into simple addition of exponents, which is precisely the insight that made logarithms — and, before calculators, slide rules — so valuable.
These laws are also why scientific notation works so cleanly: numbers written as a coefficient times a power of ten combine by adding exponents, letting you multiply astronomically large or small quantities with ease. The same rules govern compound interest, radioactive decay, population growth, and computer algorithms measured in powers of two. Exponentiation is the mathematical engine of anything that grows or shrinks by repeated proportional steps.
When exponentiation has no real answer
Not every base-and-exponent pair produces a real number. The classic case is a negative base raised to a fractional power, such as (−8)^(1/2), which asks for the square root of a negative number. No real number squared gives a negative, so the result is imaginary — it lives in the complex numbers rather than on the number line. This calculator returns 'undefined' for such inputs because it works in real numbers.
Interestingly, some fractional powers of negative bases do have real answers: (−8)^(1/3) is −2, because the cube root of a negative number is a real negative number. The distinction comes down to whether the root is odd or even. Zero raised to a negative power is another undefined case, since it demands division by zero. These edge cases are not flaws in exponentiation so much as honest signals that the operation has stepped outside the real numbers.
Frequently asked questions
What does a negative exponent mean?
A negative exponent means take the reciprocal: b⁻ⁿ equals 1 ÷ bⁿ. So 2⁻³ is 1 ÷ 2³ = 1 ÷ 8 = 0.125. It follows from keeping the exponent rules consistent as you count down past zero, where the base flips into the denominator.
Why is any number to the power of zero equal to one?
Because dividing bⁿ by b lowers the exponent by one, and continuing down to zero lands on b⁰ = 1 for any non-zero base. It is the only value that keeps the laws of exponents consistent. Zero to the zero is a special, context-dependent case.
What is a fractional exponent?
A fractional exponent is a root. b^(1/2) is the square root of b, b^(1/3) is the cube root, and b^(m/n) is the nth root of b raised to the m power. So 27^(2/3) is the cube root of 27 (which is 3), squared, giving 9.
Why does the calculator sometimes say 'undefined'?
Because the result isn't a real number. A negative base with an even fractional exponent — like the square root of a negative number — is imaginary, and zero raised to a negative power divides by zero. Both fall outside the real numbers this calculator works in.