What next?
Powers turn up in growth, geometry and algebra.
What your result means
An exponent counts how many times the base multiplies itself. Beyond whole numbers the idea extends smoothly: a negative exponent gives the reciprocal, a fractional exponent gives a root, and zero always gives 1. The scientific-notation row is the practical one once results grow past a few digits.
Why this one is different
One power comes back written several ways: the plain result, the digits in full, scientific notation, the matching root, the reciprocal and the logarithm that recovers the exponent. Results too large or too small for ordinary notation keep their precision in the scientific row instead of collapsing to zero or infinity.
Doubling thirty times gets you past a billion
2¹⁰ is a thousand, 2²⁰ is a million, 2³⁰ is a billion. Each extra step doubles everything before it, which is why exponential growth is so hard to judge by intuition — and why compound interest, viral spread and computer storage all behave in ways that feel sudden.
The same effect runs backwards. A negative exponent shrinks just as fast: 2⁻¹⁰ is under one thousandth.
How it works
For whole exponents the base is multiplied by itself that many times. For fractional exponents the calculation is defined through roots — x^(1/n) is the n-th root of x. Negative exponents are the reciprocal of the positive case, and any non-zero base to the power zero equals one.
How to use this calculator
- Enter the base — the number being raised.
- Enter the exponent, including a minus sign or a decimal if needed.
- Read the result, with scientific notation for large values.
- Use the inverse root row to check the calculation in reverse.
Formula
b = base, n = exponent. Also b⁰ = 1 for any non-zero b, and bᵐ × bⁿ = bᵐ⁺ⁿ.
Example calculation
Calculating 2¹⁰:
= 1,024
2⁻¹⁰ = 1 ÷ 1,024 ≈ 0.000977
1,024^(1/10) = 2
Frequently asked questions
Why does anything to the power zero equal one?＋
Because dividing bⁿ by bⁿ gives 1, and the exponent rule says it also gives b⁰. Both must be true, so b⁰ = 1 for any non-zero base. Zero to the power zero is left undefined by convention.
What does a fractional exponent mean?＋
It is a root. An exponent of 0.5 is the square root, one third is the cube root, and two thirds is the cube root squared. This is why x^(1/n) and the n-th root of x are the same operation.
Why do I get an error for a negative base?＋
A negative base with a fractional exponent has no real answer — the square root of a negative number is imaginary. The calculator reports this rather than returning a misleading value.
What is the difference between a power and a root?＋
They are inverse operations. Three squared is nine, and the square root of nine is three. A root can also be written as a fractional exponent, so the square root of nine is nine to the power of a half.
What does a negative exponent mean?＋
A reciprocal. Two to the power minus three is one over two cubed, which is one eighth. It never makes the result negative.
How do exponents behave when multiplied?＋
Multiplying powers of the same base adds the exponents, and dividing subtracts them. Raising a power to another power multiplies them, so two cubed squared is two to the sixth.
Where does exponential growth show up in real life?＋
Compound interest, population growth and viral spread all follow the same shape: a constant percentage applied repeatedly. The counterintuitive part is how slowly it starts and how quickly it ends.
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Assumptions & limitations
Every figure here comes from a simplified model. Keep these limits in mind when reading your result:
- Uses double-precision floating point, so results beyond about fifteen significant digits are rounded.
- Negative bases with fractional exponents have no real solution and are reported as undefined.
- Very large exponents overflow to infinity rather than returning an exact value.