Exponentiation – basics and rules
From integer powers to negative exponents
Exponentiation is one of the fundamental arithmetic operations in mathematics. The power aⁿ means: multiply the base a by itself exactly n times. The result grows exponentially: 2¹⁰ = 1024, already over a thousand, even though the base is only 2. This explosive growth explains many natural phenomena such as compound interest, population growth, and the runtime of algorithms in computer science.
Negative exponents represent the reciprocal: a⁻ⁿ = 1/aⁿ. This allows very small numbers to be expressed elegantly. In scientific notation, 0.001 is written as 10⁻³, or the speed of light as 3 × 10⁸ m/s. Negative exponents are common in physics, chemistry, and engineering. Our calculator computes 2⁻¹ = 0.5 just as correctly as 10⁻⁶ = 0.000001.
Fractional exponents connect powers and roots: a^(1/2) = √a and a^(1/3) = ³√a. In general, a^(p/q) = (ⁿ√a)ᵖ. This relationship is important for simplifying algebraic expressions and appears in calculus, probability theory, and physics. One key rule: a⁰ = 1 for all a ≠ 0. This also holds for seemingly tricky cases like 1000⁰ = 1.
Powers appear everywhere in everyday life: file sizes are measured in kilobytes (1 KB = 10³ bytes), megabytes (10⁶), and gigabytes (10⁹). With compound interest, a capital K grows at rate r after n years to K × (1 + r/100)ⁿ. Even at just 3% annual interest, a capital doubles after 24 years – a direct application of exponentiation that our calculator evaluates precisely.