What this calculator does
Powers of 2 turn up constantly in computing, from the number of values a group of bits can hold to the sizes memory and storage are sold in, because binary counting doubles with every extra bit. Working out something like 2 to the power of 5 by hand is easy enough, but the numbers grow fast, and it is easy to lose track a few steps further along.
This power of 2 calculator takes any exponent, positive, negative or zero, and returns 2 raised to that power directly, alongside a reference table listing 2⁰ through 2²⁰ for the common cases that come up in binary and memory-size contexts without retyping the exponent each time.
The formula
A positive integer exponent means repeated multiplication: 2ⁿ = 2 × 2 × …, n times. A negative exponent flips that into a fraction, 2⁻ⁿ = 1/2ⁿ, and a fractional exponent gives a root, such as 2^0.5 being the square root of 2. The reference table below simply evaluates the same formula for each whole-number exponent from 0 to 20.
| Term | Meaning |
|---|---|
| n | The exponent: how many times 2 is multiplied by itself, or the root taken if negative or fractional. |
| 2ⁿ | Two raised to the power n, doubling with every whole increase in n. |
| Bit | A single binary digit, 0 or 1; n bits can represent exactly 2ⁿ distinct values. |
The inputs explained
| Field | What to enter |
|---|---|
| Exponent (n) | The exponent to raise 2 to. Use a whole number such as 5 for "2 to the power of 5", or a negative or decimal value for a fraction or root. |
When to use it
Working out 2 to the power of 5, or any other exponent
A textbook or coding question asking for 2 to the power of 3, 4, 5 or 6 is answered directly here, with the reciprocal and the wider reference table shown alongside for context.
How many values can n bits represent?
A field of n binary bits can represent exactly 2ⁿ distinct values, from 0 up to 2ⁿ−1, which is why 8 bits (a byte) tops out at 256 values and 16 bits reaches 65,536.
Working out a memory or storage size in binary
Memory and storage capacities are conventionally sized in powers of 2, such as 1,024 bytes to a kibibyte or 1,048,576 to a mebibyte, both of which sit in the reference table here as 2¹⁰ and 2²⁰.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What is 2 to the power of 5, and other small exponents?
A range of small whole-number exponents from 3 to 8.
How large do powers of 2 get by the time n reaches 20?
A wider range of exponents, from a negative value through to 20.
Questions
What is 2 to the power of 5?
2 to the power of 5 is 32: 2 × 2 × 2 × 2 × 2. It sits directly between 2 to the power of 4 (16) and 2 to the power of 6 (64) in the doubling sequence.
What is 2 to the power of 3 and 2 to the power of 4?
2 to the power of 3 is 8, and 2 to the power of 4 is 16. Each step up the exponent doubles the previous value, since multiplying by 2 one more time is exactly what raising the exponent by 1 means.
Why are powers of 2 so common in computing?
A binary bit has exactly two states, so n bits can represent exactly 2ⁿ distinct combinations. Memory and storage sizes, addressable ranges and colour depths are all built on that same doubling, which is why 1,024, 65,536 and similar numbers appear so often.
How is this different from the general exponent calculator?
The general exponent calculator handles any base and any exponent together. This one is fixed to a base of 2 and adds a ready reference table of powers of 2, which is the specific, more convenient tool for binary and computing contexts where the base is always 2.
For any other base or exponent, see the general exponents and roots calculator. For working with logarithms, including log base 2, see the logarithm calculator.