Exponent & Power Calculator
Calculate any base raised to any exponent, including negative, fractional, and zero powers. See step-by-step working and a complete exponent rules reference. 100% private — runs in your browser.
Understanding Exponents and Powers
An exponent (or power) tells you how many times to multiply a base number by itself. For example, 3⁴ means 3 × 3 × 3 × 3 = 81. Exponents are a shorthand notation that appears throughout mathematics, science, engineering, and finance. This calculator handles all types of exponents: positive integers, negative numbers, fractions (which represent roots), and zero.
Understanding exponent rules is essential for algebra, calculus, physics, and computer science. Whether you are calculating compound interest, converting units, analyzing exponential growth, or simplifying algebraic expressions, exponent rules provide the foundation.
Essential Exponent Rules
xa × xb = xa+b — Product Rule
xa / xb = xa-b — Quotient Rule
(xa)b = xa×b — Power of a Power
x0 = 1 (for x ≠ 0) — Zero Exponent
x-n = 1/xn — Negative Exponent
x1/n = ⁿ√x — Fractional Exponent (Root)
(xy)a = xa × ya — Power of a Product
Negative Exponents
A negative exponent flips the base to the denominator. For example, 5⁻² = 1/5² = 1/25 = 0.04. This rule is consistent with the quotient rule: x³/x⁵ = x³⁻⁵ = x⁻² = 1/x². Negative exponents are common in scientific notation (10⁻³ = 0.001) and in formulas for decay, depreciation, and inverse-square laws.
Fractional Exponents
Examples
- 8^(1/3) = cube root of 8 = 2
- 16^(3/4) = (fourth root of 16)^3 = 2^3 = 8
- 27^(2/3) = (cube root of 27)^2 = 3^2 = 9
- 100^(0.5) = square root of 100 = 10
Scientific Notation and Powers of 10
Scientific notation uses powers of 10 to express very large or very small numbers compactly. The speed of light is 3 × 10⁸ m/s, the mass of an electron is 9.1 × 10⁻³¹ kg, and there are approximately 7.5 × 10¹⁸ grains of sand on Earth. Understanding exponents is essential for working with these numbers in science and engineering.
Exponents in Real Life
Compound interest uses the formula A = P(1+r)ⁿ, where n is the exponent representing the number of compounding periods. Population growth follows exponential models P = P₀eʳᵗ. Computer memory is measured in powers of 2 (2¹⁰ = 1024 = 1 KB). Sound intensity is measured in decibels using powers of 10. Understanding exponents helps you work with all of these real-world applications.
Powers of Common Bases
Powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. Powers of 3: 1, 3, 9, 27, 81, 243, 729. Powers of 10: 1, 10, 100, 1000, 10000. These sequences appear frequently in computer science, engineering, and everyday calculations.