Exponent Calculator — Powers, Roots, and Exponent Laws Explained

Exponentiation is one of the most powerful and widely-used operations in mathematics. From compound interest calculations to radioactive decay modelling, from scientific notation to logarithmic scales, exponents appear throughout science, finance, and engineering. This exponent calculator handles all cases: positive and negative exponents, fractional exponents (roots), zero exponent, and the natural base e.

What Is an Exponent?

An exponent (or power) tells you how many times to multiply a number (the base) by itself. It's written as a superscript:

aⁿ means: multiply a by itself n times

• a = base
• n = exponent (or index, or power)

Examples:

• 2³ = 2 × 2 × 2 = 8
• 5² = 5 × 5 = 25
• 10⁴ = 10 × 10 × 10 × 10 = 10,000

The 8 Laws of Exponents (Rules of Indices)

These rules apply to all real-number bases (where defined):

Law	Formula	Example
Product Rule	aᵐ × aⁿ = aᵐ⁺ⁿ	2³ × 2⁴ = 2⁷ = 128
Quotient Rule	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	3⁵ ÷ 3² = 3³ = 27
Power of a Power	(aᵐ)ⁿ = aᵐⁿ	(2²)³ = 2⁶ = 64
Power of a Product	(ab)ⁿ = aⁿbⁿ	(2×3)² = 4×9 = 36
Power of a Quotient	(a/b)ⁿ = aⁿ/bⁿ	(2/3)² = 4/9
Zero Exponent	a⁰ = 1	7⁰ = 1
Negative Exponent	a⁻ⁿ = 1/aⁿ	2⁻³ = 1/8 = 0.125
Fractional Exponent	a^(m/n) = ⁿ√(aᵐ)	8^(2/3) = ∛(8²) = ∛64 = 4

Memorising these eight rules allows you to simplify almost any expression involving exponents.

Negative Exponents

A negative exponent means "take the reciprocal":

a⁻ⁿ = 1/aⁿ

• 2⁻¹ = 1/2 = 0.5
• 5⁻² = 1/25 = 0.04
• 10⁻³ = 1/1000 = 0.001

Negative exponents appear constantly in scientific notation for very small numbers. For example, the mass of a proton is approximately 1.67 × 10⁻²⁷ kilograms.

Note: Negative exponents do NOT produce negative results (when the base is positive). A negative exponent means "one over," not "negative times."

Fractional Exponents (Roots)

Fractional exponents represent roots:

• a^(1/2) = √a (square root)
• a^(1/3) = ∛a (cube root)
• a^(1/n) = ⁿ√a (nth root)
• a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ

Examples:

• 25^(1/2) = √25 = 5
• 27^(1/3) = ∛27 = 3
• 16^(3/4) = (16^(1/4))³ = 2³ = 8

Fractional exponents unify the concepts of powers and roots into a single notation.

The Zero Exponent

Any non-zero number raised to the power of 0 equals 1.

This follows logically from the Quotient Rule: aⁿ ÷ aⁿ = a⁰. Since any number divided by itself equals 1, a⁰ = 1.

• 5⁰ = 1
• 1000⁰ = 1
• (−7)⁰ = 1

The value of 0⁰ is mathematically indeterminate — it's one of several important "indeterminate forms" in calculus. By convention it's sometimes defined as 1 in combinatorics and discrete mathematics.

Scientific Notation

Scientific notation uses powers of 10 to express very large or very small numbers compactly:

• Speed of light: 299,792,458 m/s = 2.998 × 10⁸ m/s
• Planck's constant: 6.626 × 10⁻³⁴ J·s
• Avogadro's number: 6.022 × 10²³ mol⁻¹

To convert: move the decimal point until one digit remains to the left, counting the moves. Right moves = negative exponent; left moves = positive exponent.

Euler's Number e and Natural Exponents

e ≈ 2.71828182845... is Euler's number, also called the mathematical constant or natural base. It's one of the most important numbers in mathematics, appearing naturally wherever growth or decay is continuous.

e^x is the natural exponential function. Its unique property: the derivative of e^x is e^x itself, making it the foundation of differential equations that model continuous processes.

Applications of e^x:

• Compound interest (continuously compounding): A = P × e^(rt)
• Radioactive decay: N(t) = N₀ × e^(−λt)
• Population growth: P(t) = P₀ × e^(kt)
• Normal distribution: The bell curve's peak equation contains e

Exponents in Finance

Compound interest formula: A = P × (1 + r/n)^(nt)

Where P = principal, r = annual rate, n = compounds per year, t = years.

Example: £1,000 at 5% annual interest compounded monthly for 10 years:

• A = 1000 × (1 + 0.05/12)^(12×10)
• A = 1000 × 1.004167^120
• A = 1000 × 1.6471 = £1,647.01

Related Resources

Related Calculators

• Quadratic Formula Calculator — Use exponents in quadratic equations.
• Percentage Calculator — Work with percentage growth exponents.

External Authority Resources

• Wolfram MathWorld: Exponentiation — Scientific definition and algebraic rules.
• Khan Academy: Powers and Exponents — Guide to base and power math.