Exponent Calculator

The Exponent Calculator is a free online tool that raises a base to a power. It handles negative and fractional exponents, shows the result in both decimal and scientific notation, and lists the exponent rules that govern how powers combine.

Modify the values and click the Calculate button to use.

What an Exponent Does

An exponent is repeated multiplication: 2⁵ is 2 × 2 × 2 × 2 × 2 = 32. That definition covers positive whole numbers, and everything else — zero, negatives, fractions — follows from making the rules stay consistent.

Exponent Rules Chart

Rule	Form	Example
Product	x^a × x^b = x^a+b	2³ × 2² = 2⁵ = 32
Quotient	x^a / x^b = x^a−b	2⁵ / 2² = 2³ = 8
Power of a power	(x^a)^b = x^ab	(2³)² = 2⁶ = 64
Power of a product	(xy)^a = x^ay^a	(2×3)² = 4×9 = 36
Zero exponent	x⁰ = 1	Any non-zero base
Negative exponent	x^−a = 1 / x^a	2⁻³ = 1/8 = 0.125
Fractional exponent	x^a/b = ^b√(x^a)	8^2/3 = 4

Why x⁰ = 1

Follow the quotient rule: x^a / x^a = x^a−a = x⁰. Any number divided by itself is 1, so x⁰ must be 1. The same logic gives negative exponents: x² / x⁵ = x⁻³, and computing the division directly gives 1/x³.

0⁰ is the one genuinely contested case. It is defined as 1 in combinatorics and most programming languages, and left undefined in analysis, because the limit depends on how you approach it.

Powers Chart

n	2ⁿ	3ⁿ	5ⁿ	10ⁿ
1	2	3	5	10
2	4	9	25	100
3	8	27	125	1,000
4	16	81	625	10,000
5	32	243	3,125	100,000
8	256	6,561	390,625	100,000,000
10	1,024	59,049	9,765,625	10,000,000,000

Where Exponents Appear

Compound interest, population growth, radioactive decay, sound intensity, earthquake magnitude, computer memory sizes and algorithmic complexity all rest on exponents. The reason is that exponential growth describes anything whose rate of change is proportional to its current size — which covers a remarkable share of natural and financial processes.

Frequently Asked Questions

What does a negative exponent mean?

Reciprocal, not negative result. 2⁻³ is 1/8, a positive number. Only a negative base can make the result negative.

How do I calculate a fractional exponent?

The denominator is the root and the numerator is the power: 8^2/3 means the cube root of 8, squared — 2² = 4.

Is 2³ the same as 3²?

No. 2³ = 8 and 3² = 9. Exponentiation is not commutative, unlike multiplication.