Exponent Calculator
Enter a base and an exponent to get the power, written both as an ordinary number and in scientific notation, with the three same-base exponent laws worked through in your numbers.
A power is repeated multiplication, and a fractional exponent is a root: x1/n is the n-th root of x.
Formula and calculation
Worked example
Eight to the power 1/3 is 2, the cube root, because 2 × 2 × 2 = 8. In scientific notation the same answer is 2 × 100. With the second exponent set to 2 the same-base laws work out as 81/3 × 82 = 81/3 + 2 = 82.333333 = 128.
Powers and roots are inverse operations
Raising to a power and taking a root undo each other: 2³ = 8 and the cube root of 8 is 2, which is why 81/3 = 2. The square root calculator does the same job for any root, including odd roots of negative numbers such as the cube root of −8. Logarithms go one step further: a logarithm asks which exponent produces a number, so log10(1000) = 3 for the same reason that 10³ = 1000.
Why the same-base laws hold
xa × xb means a copies of x multiplied by b more, which is a + b copies in total, so the exponents add. Dividing cancels copies, which is why xa ÷ xb = xa − b — and why xa ÷ xa = x0 = 1 for any base. Raising a power to a further power repeats the whole product b times, giving a × b copies. Those three rules are the reason you rarely need to multiply a power out by hand, and they are worked through in your own numbers above the fold.
What the results mean
- Result
- The value of the base raised to the power you entered. A whole-number exponent is repeated multiplication (210 = 1,024), a fractional exponent is a root (81/3 = 2) and a negative exponent is a reciprocal (2−3 = 0.125).
- Scientific notation
- The same answer in scientific notation, m × 10e, with the mantissa m between 1 and 10 — the form used in science and engineering when a number is far from 1.
- Same base: xa × xb
- The product law worked through in your numbers: powers of the same base multiply by adding the exponents, xa × xb = xa + b.
- Same base: xa ÷ xb
- The quotient law in your numbers: powers of the same base divide by subtracting the exponents, xa ÷ xb = xa − b, so any power divided by itself is 1.
- Power of a power: (xa)b
- The power-of-a-power law in your numbers: an exponent raised to a further exponent multiplies them, (xa)b = xa × b.
- Calculation
- The calculation written out: the repeated multiplication behind a whole-number exponent, 1 ÷ xn for a negative one, and the root behind a fraction such as 1/3.
Frequently asked questions
How do I work out 8 to the power of 1/3?
Type 8 as the base and 1/3 as the exponent and the answer is 2, because 2 × 2 × 2 = 8. A fractional exponent is a root: p/q means take the q-th root and then raise it to the power p. The square root calculator shows the same working for any root, square or not.
What does a negative exponent mean?
It means a reciprocal rather than a negative answer: 2−3 = 1 ÷ 2³ = 1 ÷ 8 = 0.125. Flipping the base under 1 is what a minus sign in the exponent does, which is why 10−2 is 0.01 and not −100.
Why is any number to the power of 0 equal to 1?
Because xn ÷ xn is 1, and the quotient law says it is xn − n = x0, so x0 must be 1 for every non-zero base. The case 00 has no such argument and is a convention — this page follows the usual one and treats it as 1.
What is 2 to the power of 10?
1,024, which is why a kilobyte is 1,024 bytes. Doubling from 1 ten times gives 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024, and the calculator reports the same answer in scientific notation as 1.024 × 103.
Why does the calculator refuse a negative base with a fractional exponent?
Because the answer would not be a real number: (−8)1/2 is the square root of −8, which needs the imaginary unit i. Odd roots of negative numbers are real — the cube root of −8 is −2 — and the square root calculator handles those directly.