Exponent & Power Calculator

Q: How do I multiply numbers with the same base?

When multiplying same-base powers, add the exponents: x^a × x^b = x^(a+b). When dividing, subtract: x^a / x^b = x^(a-b). When raising a power to a power, multiply: (x^a)^b = x^(a×b).

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.

How Exponent & Power Calculator Works

Calculate any base raised to any power including negative, fractional, and zero exponents. See step-by-step working and exponent rules. Enter your values into the form above and the calculator processes them instantly in your browser — no data is sent to any server.

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

x^a × x^b = x^a+b — Product Rule

x^a / x^b = x^a-b — Quotient Rule

(x^a)^b = x^a×b — Power of a Power

x⁰ = 1 (for x ≠ 0) — Zero Exponent

x^-n = 1/xⁿ — Negative Exponent

x^1/n = ⁿ√x — Fractional Exponent (Root)

(xy)^a = x^a × y^a — 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.

The Seven Laws of Exponents (With Worked Examples)

Almost every exponent problem reduces to one of seven rules. Learn these and you can simplify by hand before reaching for a calculator:

Product rule — same base, add the powers: a^m × aⁿ = a^m+n. So 2³ × 2⁴ = 2⁷ = 128.
Quotient rule — same base, subtract the powers: a^m ÷ aⁿ = a^m−n. So 5⁶ ÷ 5² = 5⁴ = 625.
Power of a power — multiply the powers: (a^m)ⁿ = a^m×n. So (3²)⁴ = 3⁸ = 6,561.
Power of a product — the power distributes: (ab)ⁿ = aⁿbⁿ. So (2×5)³ = 2³ × 5³ = 8 × 125 = 1,000.
Zero exponent — anything non-zero to the power 0 is 1: a⁰ = 1. It follows from the quotient rule, since aⁿ ÷ aⁿ = a⁰ and also equals 1.
Negative exponent — flip it: a⁻ⁿ = 1 ÷ aⁿ. So 4⁻² = 1/16 = 0.0625. A negative exponent never makes the answer negative.
Fractional exponent — the denominator is a root: a^m/n = the n-th root of a^m. So 8^2/3 = (cube root of 8)² = 2² = 4.

Two mistakes worth naming because they account for most wrong answers. −3² is not the same as (−3)²: without brackets the exponent binds tighter than the minus sign, so −3² = −9 while (−3)² = 9. And the product and quotient rules only work when the bases match — 2³ × 3³ cannot be collapsed by adding powers, though the power-of-a-product rule runs it the other way: 2³ × 3³ = 6³ = 216. Powers of ten drive the SI prefix system, where kilo is 10³ and micro is 10⁻⁶; the full table is maintained by NIST. Updated 2026-08-25.

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.

Frequently Asked Questions

What does a negative exponent mean?

A negative exponent means the reciprocal of the positive exponent. For example, 2^(-3) = 1/(2^3) = 1/8 = 0.125. The base is flipped to the denominator.

What is any number raised to the power of 0?

Any non-zero number raised to the power of 0 equals 1. This is a fundamental rule of exponents: x^0 = 1 for all x ≠ 0. 0^0 is typically defined as 1 by convention.

What does a fractional exponent mean?

A fractional exponent represents a root. x^(1/n) = the nth root of x. x^(m/n) = the nth root of x^m. For example, 8^(2/3) = (cube root of 8)^2 = 2^2 = 4.

Can I calculate 0 raised to a negative power?

No. 0 raised to a negative power is undefined because it would require dividing by zero (0^(-n) = 1/0^n = 1/0, which is undefined).

What is scientific notation?

Scientific notation expresses numbers as a × 10^n where 1 ≤ a < 10. For example, 5,000,000 = 5 × 10^6 and 0.003 = 3 × 10^(-3). This calculator shows results in scientific notation for very large or small numbers.

How do I multiply numbers with the same base?

When multiplying same-base powers, add the exponents: x^a × x^b = x^(a+b). When dividing, subtract: x^a / x^b = x^(a-b). When raising a power to a power, multiply: (x^a)^b = x^(a×b).