What are the basic rules of exponents?

Product rule: a^m × a^n = a^(m+n). Quotient rule: a^m ÷ a^n = a^(m−n). Power rule: (a^m)^n = a^(m×n). Zero exponent: a^0 = 1. Negative exponent: a^(−n) = 1/a^n. Fractional exponent: a^(1/n) = nth root of a.

How do you calculate negative exponents?

a^(−n) = 1/a^n. Examples: 2^(−3) = 1/8 = 0.125. 5^(−2) = 1/25 = 0.04. (2/3)^(−2) = (3/2)^2 = 9/4 = 2.25. In scientific notation: 1 × 10^(−6) = 0.000001.

What is scientific notation?

Very large or small numbers expressed as a number between 1 and 10 multiplied by a power of 10. Large: 5,400,000 = 5.4 × 10^6. Small: 0.000045 = 4.5 × 10^(−5). To multiply: multiply coefficients and add exponents.

What is the difference between 2^10 and 10^2?

2^10 = 1,024. 10^2 = 100. In computing, powers of 2 are fundamental: 2^10 = 1,024 (1 KB), 2^20 ≈ 1 million (1 MB), 2^30 ≈ 1 billion (1 GB).

What is Euler's number e?

e ≈ 2.71828 — the base of natural logarithms. Unique in that e^x equals its own derivative. Applications: continuous compound interest (A = Pe^(rt)), radioactive decay, population growth, and the normal distribution in statistics.

Exponent Calculator

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Exponent

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An exponent tells you how many times to multiply a base number by itself. 2³ = 2 × 2 × 2 = 8. The base is 2, the exponent is 3. Key rules: a⁰ = 1 for any base; a⁻ⁿ = 1/aⁿ; a^(1/n) = nth root of a. The calculator handles positive, negative, and fractional exponents, plus scientific notation for very large or small results.

Powers and exponents

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How do exponents work?

An exponent tells you how many times to multiply a number (the base) by itself. For example, 2^10 means multiply 2 by itself 10 times, which equals 1,024. Exponents are also called powers. This calculator supports any base and exponent, including negative and fractional exponents. Related: use the Log Calculator to go further, or the Scientific Calculator for a different angle.

Special Exponent Rules

Any number raised to the power of 0 equals 1. Any number raised to the power of 1 equals itself. Negative exponents create fractions: 2^(-3) = 1/8. Fractional exponents create roots: 8^(1/3) = 2 (cube root of 8).

How should you read and use the output?

The Exponent Calculator is built for people who want a fast answer without losing context. It keeps the calculation simple, shows the result clearly, and helps you understand what the number means before you use it in a real decision.

This calculator is designed to make a specific everyday calculation faster and clearer. It gives a structured result so you can compare options, check assumptions, or plan the next step with less manual work.

What are exponents?

An exponent (also called a power or index) indicates how many times a number (the base) is multiplied by itself. The expression 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. The base is 2 and the exponent is 5. Exponents appear in scientific notation, compound interest calculations, computer storage (powers of 2), physics equations, and algebra.

Notation: b^n means b raised to the power n. b is the base, n is the exponent.

What are the exponent rules?

Product rule: b^m × b^n = b^(m+n)

Example: 3^4 × 3^2 = 3^6 = 729

Quotient rule: b^m ÷ b^n = b^(m–n)

Example: 5^7 ÷ 5^3 = 5^4 = 625

Power rule: (b^m)^n = b^(m×n)

Example: (2^3)^4 = 2^12 = 4,096

Zero exponent: b^0 = 1 (for any non-zero b)

Example: 7^0 = 1, 1000^0 = 1, (–5)^0 = 1

Negative exponent: b^(–n) = 1 ÷ b^n

Example: 2^(–3) = 1/2^3 = 1/8 = 0.125

Fractional exponent: b^(1/n) = nth root of b

Example: 27^(1/3) = ∛27 = 3. 16^(1/4) = ⁴√16 = 2

Fractional exponent (m/n): b^(m/n) = (nth root of b)^m

Example: 8^(2/3) = (∛8)² = 2² = 4

What are the most common powers and their values?

2^10 = 1,024 ≈ 1 thousand | 2^20 = 1,048,576 ≈ 1 million | 2^30 ≈ 1 billion (computer storage: KB, MB, GB)
10^3 = 1,000 (kilo) | 10^6 = 1,000,000 (mega) | 10^9 (giga) | 10^12 (tera)
3^2 = 9 | 4^2 = 16 | 5^2 = 25 | 6^2 = 36 | 10^2 = 100 | 12^2 = 144
2^3 = 8 | 3^3 = 27 | 4^3 = 64 | 5^3 = 125 | 10^3 = 1,000

How is scientific notation used with exponents?

Scientific notation expresses very large or very small numbers as a number between 1 and 10 multiplied by a power of 10.

Distance to the Sun: 149,597,870 km = 1.496 × 10^8 km
Size of a hydrogen atom: 0.000000000106 m = 1.06 × 10^(–10) m
Speed of light: 299,792,458 m/s = 2.998 × 10^8 m/s

Multiplying numbers in scientific notation: multiply the coefficients and add the exponents. (3 × 10^4) × (2 × 10^6) = 6 × 10^10

Exponents in compound interest — the financial connection

The compound interest formula A = P(1 + r/n)^(nt) uses an exponent (nt) to model how money grows over time. The exponent represents the number of compounding periods, and exponential growth is why compound interest is so powerful over long periods. At 10% annual interest, $100,000 grows to $110,000 after 1 year but $672,750 after 20 years — the exponent of 20 multiplies the growth dramatically.

Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ

When multiplying powers with the same base, add the exponents. Examples: 2³ × 2⁴ = 2⁷ = 128. x⁵ × x² = x⁷. 10² × 10³ = 10⁵ = 100,000. This rule works because 2³ = 2×2×2 and 2⁴ = 2×2×2×2 — multiplying gives seven 2s multiplied together. The quotient rule is the inverse: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Example: 2⁵ ÷ 2² = 2³ = 8.

Negative exponents: a⁻ⁿ = 1/aⁿ

A negative exponent means take the reciprocal. 2⁻³ = 1/2³ = 1/8 = 0.125. 10⁻² = 1/100 = 0.01. x⁻¹ = 1/x. This is why scientific notation uses negative exponents for small numbers: 3×10⁻⁶ = 0.000003. Zero exponent: a⁰ = 1 for any non-zero base. 5⁰ = 1. (−3)⁰ = 1. This follows from the quotient rule: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰ = 1.

Fractional exponents and roots: a^(1/n) = ⁿ√a

A fractional exponent represents a root. a^(1/2) = √a. a^(1/3) = ∛a. a^(m/n) = (ⁿ√a)ᵐ. Examples: 8^(1/3) = ∛8 = 2. 16^(3/4) = (⁴√16)³ = 2³ = 8. 25^(1/2) = √25 = 5. 27^(2/3) = (∛27)² = 3² = 9. Fractional exponents allow root calculations using exponent rules — the same product and quotient rules apply.

Frequently asked questions about exponents

What is any number to the power of 1? Any base to the power of 1 equals itself. 7^1 = 7. 1,000^1 = 1,000.

What is 0 to the power of 0? 0^0 is mathematically indeterminate — it has no single well-defined value. In most computing and mathematical contexts, it is treated as 1 by convention, but this is context-dependent.

What is a negative base raised to an exponent? A negative base raised to an even exponent gives a positive result. A negative base raised to an odd exponent gives a negative result. (–3)^2 = 9. (–3)^3 = –27. — see also: log calculator

Powers of 2 — computing reference table

Power	Value	Computing Equivalent
2^0	1	1 bit has 1 state
2^1	2	1 bit has 2 values (0 or 1)
2^8	256	1 byte — 256 possible values
2^10	1,024	1 kilobyte (KB)
2^16	65,536	16-bit colour — 65,536 colours
2^20	1,048,576	1 megabyte (MB)
2^24	16,777,216	24-bit true colour — 16.7M colours
2^30	1,073,741,824	1 gigabyte (GB)
2^32	4,294,967,296	Max 32-bit integer; IPv4 addresses
2^40	~1 trillion	1 terabyte (TB)
2^64	~18.4 quintillion	Max 64-bit integer

Powers of 10 — scientific notation reference

Power	Value	SI Prefix	Example
10^-12	0.000000000001	pico (p)	Picosecond
10^-9	0.000000001	nano (n)	Nanometre (nm)
10^-6	0.000001	micro (μ)	Micrometre
10^-3	0.001	milli (m)	Millimetre
10^0	1	—	Base unit
10^3	1,000	kilo (k)	Kilogram, kilometre
10^6	1,000,000	mega (M)	Megabyte, megawatt
10^9	1,000,000,000	giga (G)	Gigabyte, gigahertz
10^12	1,000,000,000,000	tera (T)	Terabyte, terahertz

Sources & References

Sources: Stewart, J. Calculus: Early Transcendentals (8th ed); NIST Digital Library of Mathematical Functions, Chapter 4 — Elementary Functions.

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Formula reviewed by Mayra · Methodology · Last reviewed: June 2026