Scientific Calculator

The square root of x is a value y such that y × y = x. For example, √9 = 3 because 3 × 3 = 9. Every positive number has two square roots: a positive one (√x) and a negative one (−√x).

An exponent (power) tells you how many times to multiply a base by itself. For example, 2³ = 2 × 2 × 2 = 8. Any number to the power of 0 equals 1 (e.g., 5⁰ = 1).

A negative exponent means the reciprocal: b⁻ⁿ = 1/bⁿ. For example, 2⁻³ = 1/8 = 0.125. Negative exponents are common in scientific notation, e.g., 1 nanometer = 10⁻⁹ meters.

Key rules: bᵐ × bⁿ = bᵐ⁺ⁿ (multiply same base: add exponents); bᵐ ÷ bⁿ = bᵐ⁻ⁿ (divide: subtract); (bᵐ)ⁿ = bᵐⁿ (power of power: multiply); (ab)ⁿ = aⁿbⁿ (power of product).

n! = n × (n-1) × … × 2 × 1. For example, 5! = 120. By definition, 0! = 1. Factorials grow extremely fast — 20! is already over 2.4 quintillion.

Permutations count ordered arrangements (ABC ≠ BAC). Combinations count unordered selections (ABC = BAC). nPr = n!/(n-r)!, nCr = n!/(r!(n-r)!).

logᵦ(x) = y means bʸ = x. Logarithm is the inverse of exponentiation. For example, log₂(8) = 3 because 2³ = 8.

The natural logarithm (ln) uses base e (≈ 2.71828). It appears in calculus, physics, and finance (e.g., continuous compounding). ln(e) = 1 by definition.