Exponent Calculator

This exponent calculator works out the power of a number — aⁿ — for any base and exponent, including negative and fractional powers, and shows the result with scientific notation and a step-by-step explanation. Beyond a single exponent calculation, the guide below covers the full rules of exponents: adding exponents, subtracting exponents, multiplying exponents and simplifying exponential expressions, so you can both compute and understand.

What Is an Exponent?

An exponent (also called a power) is repeated multiplication of a base: aⁿ = a × a × … × a (n times). Here a is the base and n is the exponent. So the answer to "what is an exponent" is simply a shorthand for repeated multiplication: 2³ = 2×2×2 = 8 and 10⁶ = 1,000,000. Powers like these are the foundation of scientific notation, logarithms, exponential growth and roots, which is why mastering them unlocks so much of algebra.

The Rules of Exponents

The rules of exponents (also called the laws of exponents) let you simplify any exponential expression without expanding it fully:

• a⁰ = 1 — Zero exponent (for a ≠ 0). Example: 7⁰ = 1.
• a¹ = a — The first power is always the base itself.
• a⁻ⁿ = 1/aⁿ — Negative exponent. Example: 2⁻³ = 1/8 = 0.125.
• aⁿ × aᵐ = aⁿ⁺ᵐ — Same base multiplication: add exponents.
• aⁿ ÷ aᵐ = aⁿ⁻ᵐ — Same base division: subtract exponents.
• (aⁿ)ᵐ = aⁿˣᵐ — Power of a power: multiply exponents.
• (a×b)ⁿ = aⁿ × bⁿ — Power of a product.
• a^(1/n) = ⁿ√a — A fractional exponent equals a root: 8^(1/3) = ∛8 = 2.

Adding and Subtracting Exponents

Adding exponents as numbers — the way you add ordinary terms — only works when the base and the exponent are identical. These are called like terms: 3² + 3² = 2 × 3² = 18. You cannot simply add the exponents themselves; 2³ + 2² is not 2⁵. The same applies to subtracting exponents: 5³ − 5³ = 0, but 5³ − 5² must be evaluated separately (125 − 25 = 100). Note the common confusion — adding exponential expressions combines like terms, while the rule aⁿ × aᵐ = aⁿ⁺ᵐ adds the exponents only during multiplication.

Multiplying and Dividing Exponents

Multiplying exponents with the same base means you add the powers, and dividing means you subtract them. This is the heart of working with exponential expressions:

• Same base, multiplying: aⁿ × aᵐ = aⁿ⁺ᵐ. Example: 3² × 3⁴ = 3⁶ = 729. This is also the rule for multiplying powers and multiplying exponential expressions.
• Same base, dividing: aⁿ ÷ aᵐ = aⁿ⁻ᵐ. Example: 7⁵ ÷ 7² = 7³ = 343 — the core of dividing exponential expressions.
• Different base, same exponent: aⁿ × bⁿ = (a×b)ⁿ. Example: 2³ × 5³ = 10³ = 1000.
• Different base and exponent: calculate each power, then combine. 2³ × 3² = 8 × 9 = 72.

Negative and Fractional Exponents

A negative exponent means the reciprocal: a⁻ⁿ = 1/aⁿ. Example: 5⁻² = 1/25 = 0.04. A fractional exponent means a root: a^(m/n) = ⁿ√(aᵐ). Example: 16^(3/4) = (⁴√16)³ = 2³ = 8. This exponent calculator handles all of these cases automatically, so you can check any power of a number instantly.

Powers of 10 and Scientific Notation

Powers of 10 are the basis of scientific notation, used to write very large or very small numbers compactly:

Where Powers Are Used in Real Life

• Compound interest: A = P(1 + r)ⁿ uses a power to model growth over n periods.
• Computing: Data sizes are powers of 2 (1 KB = 2¹⁰ bytes, 1 MB = 2²⁰ bytes).
• Science: Scientific notation expresses atomic and astronomical scales.
• Geometry: Area uses squares (a²) and volume uses cubes (a³).

Enter any base and exponent above to compute the power of a number, complete with scientific notation and a worked explanation. For more answers on negative powers, fractional exponents and the rules of exponents, see the frequently asked questions below.