Exponentiation Calculator
Powers, roots, negative & fractional exponents — with step-by-step working
Calculate bn
Supports decimals, negatives, and fractions like 1/3
Any real number
Integer, decimal, or fraction
Formula: b^n = b × b × … (n times) · Results ≥ 10¹⁵ shown in scientific notation automatically
Nth Root Calculator
Find any root: square root, cube root, 4th root, and beyond
The radicand
Which root to take
Formula: ⁿ√a = a^(1/n)
Find the Exponent
Given base and result, solve for the exponent using logarithms
Must be positive, not 1
Must be positive
Solving: b^x = y → x = log(y) / log(b)
Powers Table
See all powers from exponent 0 to 10 for any base
Worked Examples
4×2 = 8
8×2 = 16
16×2 = 32
32×2 = 64
64×2 = 128
128×2 = 256
256×2 = 512
512×2 = 1,024
Fractional rule: a^(p/q) = ⁿ√a
∛27 = ?
3 × 3 × 3 = 27 ✓
Result = 3
2^(-3) = 1 / 2^3
= 1 / (2×2×2)
= 1 / 8
= 0.125
Laws of Exponents — Quick Reference
Quotient: a^m / a^n = a^(m−n)
Power²: (a^m)^n = a^(m×n)
Negative: a^(-n) = 1/a^n
Fractional: a^(p/q) = ⁿ√(a^p)
Understanding Exponentiation
Exponentiation — also called "raising to a power" — is a fundamental mathematical operation that compresses repeated multiplication into a compact notation. When we write bn, we mean the base b multiplied by itself n times. For instance, 54 = 5 × 5 × 5 × 5 = 625. The operation is ubiquitous: it governs compound interest, population growth, radioactive decay, digital storage capacities, and the speed of algorithms.
The two key components of any exponential expression are the base (the number being multiplied) and the exponent or power (how many times it is multiplied). While integer exponents are the most intuitive, the concept extends naturally to negative exponents (reciprocals), fractional exponents (roots), and even irrational or complex exponents through more advanced mathematics.
Positive Integer Exponents
A positive integer exponent n on base b simply means b multiplied by itself n times. The result grows rapidly, especially for bases greater than 1. This is the basis of exponential growth observed in nature, technology, and finance. The classic example is doubling: starting with 1 and doubling 10 times gives 1,024 — a fact central to understanding binary computing, where data is stored in powers of 2.
Negative Exponents
A negative exponent indicates a reciprocal. The rule is: a−n = 1 / an. So 10−3 = 1 / 1000 = 0.001. Negative exponents are the basis of scientific notation for small numbers — the mass of an electron is approximately 9.11 × 10−31 kilograms. They also arise naturally in unit conversions and probability calculations.
Fractional Exponents and Roots
A fractional exponent p/q combines powers and roots: a(p/q) is the q-th root of ap. The most common cases are square roots (exponent 1/2) and cube roots (exponent 1/3). For example, 64(1/2) = √64 = 8 and 125(1/3) = ∛125 = 5. Fractional exponents unify roots and powers under a single algebraic system, making it easy to apply the standard laws of exponents to root operations as well.
Laws of Exponents
The seven fundamental laws of exponents allow simplification of complex expressions:
- Product rule: am × an = am+n — multiply same bases by adding exponents.
- Quotient rule: am / an = am−n — divide same bases by subtracting exponents.
- Power of a power: (am)n = am×n — raise a power to another power by multiplying.
- Power of a product: (ab)n = an × bn — distribute the exponent across a product.
- Zero exponent: a0 = 1 for any non-zero a.
- Negative exponent: a−n = 1 / an.
- Fractional exponent: a(p/q) = q-th root of ap.
Real-World Applications
Exponentiation appears across virtually every field of science and daily life. Compound interest uses the formula A = P(1 + r)t, where t is the number of compounding periods. The Richter scale for earthquakes and the decibel scale for sound are both logarithmic — meaning a difference of 1 on the scale represents a tenfold (101) increase in magnitude. Computer memory and file sizes are expressed in powers of 2 (kilobytes = 210 bytes, megabytes = 220 bytes). Population biology models exponential growth and decay with bt expressions where the base represents the growth factor.
Scientific Notation
Scientific notation writes any number as a coefficient between 1 and 10 multiplied by a power of 10. The number 6,022,000,000,000,000,000,000,000 (Avogadro's number) becomes 6.022 × 1023 — far more readable and easier to manipulate algebraically. Very small numbers like 0.000000000167 become 1.67 × 10−10. When multiplying two numbers in scientific notation, simply add the powers: (2 × 103) × (4 × 105) = 8 × 108.