How to Calculate Exponents
This calculator performs exponentiation of any number, showing the result in both decimal notation and scientific notation.
The Definition of a Power
Exponentiation is the operation that takes two numbers -- the base (a) and the exponent (n) -- and produces their result a^n. When the exponent is a positive integer, the power is the product of the base multiplied by itself n times.
Special Cases
- Exponent 0: a^0 = 1 for any a not equal to 0. This convention ensures consistency of the properties of powers.
- Exponent 1: a^1 = a. Any number raised to 1 remains unchanged.
- Negative exponent: a^(-n) = 1/a^n. Indicates the reciprocal of the positive power.
- Fractional exponent: a^(1/n) = nth root of a. Powers with fractional exponents are roots.
- Base 0: 0^n = 0 for n > 0. The case 0^0 is an indeterminate form.
Properties of Exponents
The properties of exponents are fundamental rules of algebra:
- Product: a^m x a^n = a^(m+n) -- add the exponents
- Quotient: a^m / a^n = a^(m-n) -- subtract the exponents
- Power of a power: (a^m)^n = a^(m x n) -- multiply the exponents
- Power of a product: (a x b)^n = a^n x b^n -- the power distributes
Scientific Notation
Scientific notation is a compact way to write very large or very small numbers. The number is expressed as c x 10^n, where 1 <= c < 10 and n is an integer. For example, the speed of light: 300,000,000 m/s = 3 x 10^8 m/s.
Powers in Real Life
Powers describe phenomena of exponential growth: bacterial growth (doubling), compound interest in finance, viral spread on social media. Understanding powers is essential for grasping how small changes can produce enormous effects over time.