An exponent counts how many times a base multiplies itself: 2⁵ is 32. The rules follow from that counting. Multiplying powers of the same base adds the exponents, because you are simply concatenating the lists of factors, and dividing subtracts them.
The rules then extend the notation beyond whole numbers. A zero exponent gives one, because dividing a power by itself subtracts to zero and equals one. A negative exponent gives the reciprocal, since 2⁻³ is 1 ÷ 2³ = 0.125. A fractional exponent gives a root, so 8^(1/3) is the cube root, 2. None of these are separate conventions bolted on; each is what the addition rule forces if it is to hold everywhere. This page shows which reading applies to your input.
The formula
b- The base being raised
n- The exponent or index
b<sup>0</sup>- Always 1 for any non-zero base
b<sup>1/n</sup>- The nth root of the base
How it works, step by step
- Enter the base.
- Enter the exponent, which may be negative or fractional.
- The gauge shows the result.
- The reciprocal and root readings appear below where they apply.
Worked examples
2 to the power 10
2¹⁰ = 1,024, which is why a kilobyte was 1,024 bytes rather than 1,000. Ten doublings is close enough to a factor of a thousand that the two got conflated for decades.
Negative and fractional exponents
2⁻³ = 1 ÷ 2³ = 0.125, the reciprocal. And 8^(1/3) = 2, the cube root. Both follow from the rule that multiplying powers adds their exponents.
How to read your score
Frequently asked questions
Why does anything to the power zero equal one?
Because dividing a power by itself subtracts the exponents to zero, and any value divided by itself is one. For the rule to hold consistently, b⁰ must be one for every non-zero base.
What does a negative exponent mean?
The reciprocal of the positive power. 2⁻³ is 1 ÷ 2³ = 0.125. It follows from the subtraction rule once the exponent goes below zero.
How do fractional exponents relate to roots?
b^(1/n) is the nth root of b. So b^(2/3) is the cube root squared. The notation unifies powers and roots into one operation.
What is 0 to the power 0?
It is conventionally taken as one, which makes power series and the binomial theorem work cleanly, though the limit is genuinely ambiguous depending on how you approach it.
Powers of 2 and the exponent rules
| Exponent n | 2<sup>n</sup> | Reading | Reciprocal 2<sup>−n</sup> |
|---|---|---|---|
| -3 | 0.125 | a reciprocal | 8 |
| -1 | 0.5 | a reciprocal | 2 |
| 0 | 1 | always 1 | 1 |
| 1/2 | 1.41421 | a root | 0.707107 |
| 1 | 2 | repeated multiplication | 0.5 |
| 5 | 32 | repeated multiplication | 0.03125 |
| 10 | 1024 | repeated multiplication | 0.000976562 |
| 20 | 1.04858e+06 | repeated multiplication | 9.53674e-07 |
Every row follows from the single rule that multiplying powers adds their exponents.