Why does anything to the power of zero equal 1?

It follows from dividing equal powers: a³ ÷ a³ equals a⁰ by the subtraction law, and any number divided by itself is 1.

What does a negative exponent mean?

A reciprocal, not a negative result. 2⁻³ is 1 divided by 2³, which is 0.125.

What is a fractional exponent?

A root. Raising to the power of 1/2 is a square root, and 1/3 is a cube root, so 8^(1/3) equals 2.

Is −3² equal to 9?

No, it is −9. The exponent applies before the minus sign. You need brackets, (−3)², to get 9.

Exponent Calculator - Powers and Indices

Exponent Calculator

An exponent tells you how many times to multiply a number by itself. The laws of indices then let you combine powers without expanding them.

Formula

aᵐ × aⁿ = aᵐ⁺ⁿ · aᵐ ÷ aⁿ = aᵐ⁻ⁿ · (aᵐ)ⁿ = aᵐⁿ

a⁰ = 1 for any non-zero a
a⁻ⁿ = 1 ÷ aⁿ
a^(1/n) is the nth root of a

2³ × 2⁴ = 2⁷ = 128

The Laws of Indices

Multiplying: a^m × aⁿ = a^m+n — add the powers
Dividing: a^m ÷ aⁿ = a^m−n — subtract them
Power of a power: (a^m)ⁿ = a^mn — multiply them

These only apply when the base is the same. 2³ × 3² cannot be combined this way.

Zero, Negative and Fractional Powers

a⁰ = 1 for any non-zero a. It follows from the division law: a³ ÷ a³ = a⁰, and anything divided by itself is 1.
a⁻ⁿ = 1 ÷ aⁿ. A negative power means a reciprocal, not a negative answer: 2⁻³ = 1/8.
a^1/n is the nth root. 8^1/3 = 2.

How Fast Powers Grow

Exponential growth outpaces intuition badly. Doubling something 20 times multiplies it by over a million; 30 times exceeds a billion.

The classic illustration is the chessboard: one grain on the first square, doubling each time, reaches roughly 18 quintillion grains by the 64th — more wheat than has ever been grown.

Note the order of operations too: −3² is −9, because the power binds tighter than the minus sign, while (−3)² is 9.

Quick Reference

Powers of two, which underpin computing.
Power	Value	Also known as
2⁸	256	One byte of values
2¹⁰	1,024	1 KiB
2¹⁶	65,536	16-bit range
2²⁰	1,048,576	1 MiB
2³²	4,294,967,296	IPv4 address space

Frequently Asked Questions

Why does anything to the power of zero equal 1?: It follows from dividing equal powers: a³ ÷ a³ equals a⁰ by the subtraction law, and any number divided by itself is 1.
What does a negative exponent mean?: A reciprocal, not a negative result. 2⁻³ is 1 divided by 2³, which is 0.125.
What is a fractional exponent?: A root. Raising to the power of 1/2 is a square root, and 1/3 is a cube root, so 8^(1/3) equals 2.
Is −3² equal to 9?: No, it is −9. The exponent applies before the minus sign. You need brackets, (−3)², to get 9.