What this calculator does
An exponent counts repeated multiplication, but the idea extends well beyond whole numbers. Fractional exponents are roots, negative exponents are reciprocals, and a zero exponent always gives one. That consistency is what makes exponents work across the whole number line rather than just the counting numbers.
This calculator handles all of those cases from the same two inputs, and reports the result in scientific notation as well, since powers grow large quickly enough that plain decimal notation stops being readable.
The formula
Raising to the power n multiplies the base by itself n times for whole numbers. A fractional exponent 1/n is the nth root, since taking that root n times returns the original. A negative exponent is the reciprocal of the positive power, which follows from the rule that dividing powers subtracts exponents.
| Term | Meaning |
|---|---|
| Base | The number being raised to a power. |
| Exponent | The power: how many times the base multiplies itself. |
| nth root | The inverse operation: a^(1/n). |
| Reciprocal power | a⁻ⁿ = 1 ÷ aⁿ. |
The inputs explained
| Field | What to enter |
|---|---|
| Base | The base. Negative bases with fractional exponents give undefined results in real arithmetic. |
| Exponent | The exponent. Fractions give roots, negatives give reciprocals, and zero always gives one. |
When to use it
Calculating compound growth
Growth over n periods raises the multiplier to the nth power. It is the same arithmetic that underlies compound interest.
Working with very large or small numbers
The scientific notation output keeps results readable when powers grow beyond what plain decimals convey. 2^64 means little as a digit string, and a great deal as 1.845 × 10¹⁹.
Finding roots
The nth root line inverts the power. Square and cube roots are given separately since they come up most often.
Understanding binary scales
Powers of two describe memory, addressing and doubling processes. Each additional power doubles the total, which is why capacity jumps in the sequence it does.
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.
Powers of 2
Doubling repeatedly, which underlies binary computing and every doubling process.
| Exponent | aⁿ | Scientific notation | Reciprocal power a⁻ⁿ |
|---|---|---|---|
| 2^8 | 256.000 | 2.560000e+2 | 0.0039 |
| 2^10 | 1,024.00 | 1.024000e+3 | 0.000977 |
| 2^16 | 65,536.00 | 6.553600e+4 | 0.000015 |
| 2^20 | 1,048,576.00 | 1.048576e+6 | 0.000001 |
| 2^32 | 4,294,967,296.00 | 4.294967e+9 | 2.3283e-10 |
| 2^64 | 1.8447e+19 | 1.844674e+19 | 5.4210e-20 |
The same exponent across different bases
Cubing a range of bases.
| Base | aⁿ | nth root of a | Square root · cube root |
|---|---|---|---|
| 2³ | 8.000 | 1.260 | 1.414 · 1.260 |
| 3³ | 27.000 | 1.442 | 1.732 · 1.442 |
| 5³ | 125.000 | 1.710 | 2.236 · 1.710 |
| 10³ | 1,000.00 | 2.154 | 3.162 · 2.154 |
| 12³ | 1,728.00 | 2.289 | 3.464 · 2.289 |
| 25³ | 15,625.00 | 2.924 | 5.000 · 2.924 |
Questions
What does a negative exponent mean?
The reciprocal of the positive power: a⁻ⁿ equals 1 ÷ aⁿ. It follows from the rule that dividing powers subtracts exponents, which forces negative exponents to behave this way for consistency.
Why is anything to the power zero equal to one?
Because dividing a power by itself gives a⁰, and any non-zero number divided by itself is one. Zero to the power zero is left undefined, since different limiting arguments give different answers.
How do fractional exponents work?
a^(1/n) is the nth root of a. So a^(2/3) is the cube root of a, squared. This is what allows exponents to extend smoothly to all real numbers rather than just integers.
What is scientific notation?
A number written as a value between 1 and 10 multiplied by a power of ten. It keeps very large and very small numbers readable and makes their order of magnitude immediately visible.
Can I raise a negative number to a fractional power?
Not within real numbers, in general. The square root of a negative number is undefined in the reals, though it exists in the complex numbers. The calculator will return an undefined result for such cases.
For the inverse operation, see the logarithm calculator. For very large or small numbers, use scientific notation.