Mathematics
Exponent and power calculator
Base and exponent: the result appears with the working as you would write it on the board, a chart of y = aˣ and a ruler showing how big the number really is.
How powers work
A power with a positive whole exponent is repeated multiplication: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. With exponent zero the result is always 1, because aⁿ / aⁿ = a⁰. A negative exponent takes the reciprocal: 10⁻³ = 1/10³ = 0.001.
A fractional exponent is a root: a^(1/2) is the square root, a^(1/3) the cube root, and 16^(3/4) = (⁴√16)³ = 2³ = 8. That is why a negative base with exponent 1/2 has no real result while with 1/3 it does: the cube root of −8 is −2.
Exponential growth always surprises: 2¹⁰ passes a thousand, 2²⁰ a million and 2¹⁰⁰ has 31 digits. The order-of-magnitude ruler helps picture it, and the chart shows how a base just above 1, such as 1.05, grows slowly at first and then faster and faster.
Common mistakes
- Confusing −2² with (−2)²: the first is −4, the second 4.
- Multiplying base and exponent: 3⁴ is 81, not 12.
- Thinking a negative exponent gives a negative result: 2⁻³ is 0.125.
Frequently asked questions
What is 0 to the power 0?
It is an indeterminate form: algebra often sets it to 1 for convenience, but as a limit it can tend to different values.
How do you raise a number to a decimal power?
Write the exponent as a fraction: a^0.75 = a^(3/4) = (⁴√a)³. For irrational exponents, aⁿ = e^(n·ln a).
What are the laws of exponents?
aᵐ · aⁿ = aᵐ⁺ⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ and (a·b)ⁿ = aⁿ·bⁿ. They hold for positive bases and, with integer exponents, for any base.
How this calculation works
The value is |a|ⁿ computed in floating point, negative when the base is negative and the exponent is an odd integer or a fraction with odd numerator and odd denominator. When the result is beyond what a double-precision number holds, it is shown in scientific notation from log₁₀|aⁿ| = n·log₁₀|a|.