What is an exponent and how do you calculate it?

An exponent (also called a power) tells you how many times to multiply a number (the base) by itself. The notation bⁿ means b multiplied by itself n times. For example, 2³ = 2 × 2 × 2 = 8. The base is 2, the exponent is 3, and the result is 8. Key terminology: • Base: the number being multiplied (2 in 2³) • Exponent / Power / Index: how many times to multiply (3 in 2³) • Result / Power: the answer (8 in 2³) • b² is called 'b squared' (area of a square with side b) • b³ is called 'b cubed' (volume of a cube with side b) • For higher powers: b⁴ = 'b to the fourth', b⁵ = 'b to the fifth', etc. • Calculating: enter base and exponent in this calculator for instant results

What happens with negative exponents?

A negative exponent means take the reciprocal (flip it). The rule is: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8 = 0.125. A negative exponent does NOT make the result negative — it makes it a fraction. The result can still be positive or negative depending on the base. Negative exponent examples: • 2⁻¹ = 1/2 = 0.5 • 2⁻² = 1/4 = 0.25 • 2⁻³ = 1/8 = 0.125 • 10⁻³ = 1/1000 = 0.001 • 5⁻² = 1/25 = 0.04 • (−3)⁻² = 1/(−3)² = 1/9 ≈ 0.111 (positive, because base is squared) • Negative exponents are used in scientific notation for very small numbers: e.g. 6.7 × 10⁻¹¹

What does a fractional exponent mean?

A fractional exponent represents a root. The rule is: a^(1/n) = the nth root of a. So a^(1/2) = √a (square root), a^(1/3) = ∛a (cube root), and a^(1/4) = ⁴√a (fourth root). More generally, a^(m/n) = (nth root of a)^m. Fractional exponent examples: • 4^(1/2) = √4 = 2 • 8^(1/3) = ∛8 = 2 • 16^(1/4) = ⁴√16 = 2 • 9^(0.5) = √9 = 3 • 27^(2/3) = (∛27)² = 3² = 9 • 4^(1.5) = 4^(3/2) = (√4)³ = 2³ = 8 • Note: negative bases with fractional exponents produce complex (imaginary) numbers in the real number system

What are the main laws of exponents?

The laws of exponents are rules for simplifying expressions with powers. They work for any base a (where a ≠ 0) and any exponents m and n. The six essential laws: • Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ (add exponents when multiplying same base) • Quotient rule: aᵐ / aⁿ = aᵐ⁻ⁿ (subtract exponents when dividing same base) • Power rule: (aᵐ)ⁿ = aᵐⁿ (multiply exponents when raising a power to a power) • Zero exponent: a⁰ = 1 for any a ≠ 0 • Negative exponent: a⁻ⁿ = 1/aⁿ • Product to a power: (ab)ⁿ = aⁿbⁿ Common mistakes to avoid: • aᵐ × bⁿ ≠ (ab)ᵐ⁺ⁿ (product rule only applies to same base) • (a + b)ⁿ ≠ aⁿ + bⁿ (you must expand using binomial theorem) • (−a)² = a² but −a² = −(a²) — brackets matter

Why does any number to the power of 0 equal 1?

The rule a⁰ = 1 (for a ≠ 0) follows logically from the quotient rule of exponents. Using the quotient rule: aⁿ / aⁿ = aⁿ⁻ⁿ = a⁰. But any number divided by itself equals 1, so a⁰ must equal 1. This is not just a convention — it is forced by the requirement that the laws of exponents remain consistent. Intuitive explanation via pattern: • 2⁴ = 16 • 2³ = 8 (divided by 2) • 2² = 4 (divided by 2) • 2¹ = 2 (divided by 2) • 2⁰ = 1 (divided by 2) ✓ • 2⁻¹ = 0.5 (divided by 2) ✓ • The pattern of dividing by the base each time requires 2⁰ = 1 • 0⁰ is a special case: mathematically undefined in analysis, but defined as 1 by convention in combinatorics and power series (this calculator uses 1)

Exponent Calculator — Powers, Roots & Laws of Exponents

The Exponent Calculator computes bⁿ for any base b and exponent n — including negative bases, negative exponents (producing reciprocals), fractional exponents (producing roots), and zero. Enter any base and any exponent: the result is shown instantly with scientific notation for very large or very s...

EXPONENT CALCULATOR

2¹⁰ = 1,024

Base

can be negative or decimal

Exponent (Power)

negative, zero, or decimal ok

RESULT

1,024

STEP-BY-STEP

2¹⁰ = 1,024

QUICK EXAMPLES

LIVE DIAGRAM

2¹⁰ =

1,024

MAGNITUDE (log scale)

1,024

POWERS OF 2

2⁰

2¹

2²

2³

2⁴

2⁵

2⁶

2⁷

128

2⁸

256

2⁹

512

2¹⁰

1,024

BASE

EXPONENT

RESULT

1,024

TYPE

Positive exp

Live diagram · updates as you type

Created with❤️byeaglecalculator.com

LAWS OF EXPONENTS

APPLIED WITH BASE = 2

LAW	EXAMPLE	RESULT
aᵐ × aⁿ = aᵐ⁺ⁿ	2⁴ × 2⁶	2¹⁰ = 1,024
aᵐ / aⁿ = aᵐ⁻ⁿ	2¹⁰ / 2⁴	2⁶ = 64
(aᵐ)ⁿ = aᵐⁿ	(2⁴)⁶	2²⁴ = 16,777,216
a⁻ⁿ = 1/aⁿ	2⁻⁴	1/2⁴ = 0.0625
(ab)ⁿ = aⁿbⁿ	(2×2)⁴	16×16 = 256
a⁰ = 1	2⁰	1

HOW TO USE

1
Enter the base — any real number including negatives and decimals. The base is the number being raised to a power.
2
Enter the exponent (power). Negative exponents give reciprocals (e.g. 2⁻³ = 1/8). Fractional exponents give roots (e.g. 4^0.5 = √4 = 2). Zero always gives 1.
3
The result appears instantly with all relevant forms: decimal, scientific notation for large/small numbers, fraction for negative exponents, and root notation for fractional exponents.
4
Read the step-by-step explanation to understand how the result is derived — especially useful for negative and fractional exponents.
5
Check the Laws of Exponents table — each law is illustrated with your specific base so the rules are applied to real numbers, not abstract symbols.

WORKED EXAMPLE

Example 1: 2¹⁰ = 2×2×2×2×2×2×2×2×2×2 = 1,024. Example 2 (negative): 2⁻³ = 1/2³ = 1/8 = 0.125. Example 3 (root): 2^0.5 = √2 ≈ 1.41421356. Example 4 (negative base even power): (−2)⁴ = 16 (positive). Example 5 (negative base odd power): (−2)³ = −8 (negative). Example 6: 5⁰ = 1. Example 7: 10³ = 1,000 = 1 × 10³.

REFERENCE FORMULAS

EXPONENT LAWS REFERENCE

NAME	FORMULA	DESCRIPTION
Product rule	aᵐ × aⁿ = aᵐ⁺ⁿ	Add exponents when multiplying same base
Quotient rule	aᵐ / aⁿ = aᵐ⁻ⁿ	Subtract exponents when dividing same base
Power of a power	(aᵐ)ⁿ = aᵐⁿ	Multiply exponents when raising power to a power
Zero exponent	a⁰ = 1	Any non-zero base to the power 0 equals 1
Negative exponent	a⁻ⁿ = 1/aⁿ	Negative exponent means reciprocal
Fractional exponent	a^(1/n) = ⁿ√a	Fractional exponent means nth root
Product to a power	(ab)ⁿ = aⁿbⁿ	Distribute exponent to each factor

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Last updated: April 28, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.