How Exponents Work: Powers and the Laws of Exponents

Exponents are repeated multiplication

Writing 2^4 is shorthand for 2 × 2 × 2 × 2. The base is the number being multiplied (2), and the exponent, or power, is how many copies you multiply (4). So 2^4 = 16 and 10^3 = 1,000. The notation exists because repeated multiplication grows fast and would be tedious to write out — 2^10 is already 1,024.

Two special names come up a lot: raising to the power 2 is called squaring (because 5^2 is the area of a square with side 5), and raising to the power 3 is called cubing (the volume of a cube). Everything else is just read as "to the power of" — 7^5 is "seven to the power of five."

The laws of exponents

Three core rules do most of the work. Multiplying same-base powers adds the exponents: 2^3 × 2^2 = 2^(3+2) = 2^5, because you are just multiplying five 2s in a row. Dividing subtracts them: 2^5 ÷ 2^2 = 2^(5−2) = 2^3, since two of the factors cancel. Raising a power to a power multiplies them: (2^3)^2 = 2^(3×2) = 2^6, because you take three 2s and repeat that twice.

The zero-exponent rule falls out of the division rule. Since 2^3 ÷ 2^3 = 2^(3−3) = 2^0, and any number divided by itself is 1, we must have 2^0 = 1. This holds for every nonzero base: 5^0, 100^0, and (−3)^0 all equal 1.

Negative and fractional exponents

A negative exponent means take the reciprocal: 2^−1 = 1 ÷ 2 = 0.5, and 2^−3 = 1 ÷ 2^3 = 1 ÷ 8 = 0.125. Counting down the powers of 2 — 8, 4, 2, 1 — each step halves, so continuing past 2^0 = 1 lands naturally on ½, ¼, and so on. Negative exponents keep that halving pattern consistent below zero.

A fractional exponent means a root. The denominator gives the root and the numerator gives the power: 9^(1/2) is the square root of 9, which is 3, and 8^(1/3) is the cube root of 8, which is 2. Combining them, 8^(2/3) = (8^(1/3))^2 = 2^2 = 4. This is why roots and powers are really the same operation seen from two sides.

Frequently asked questions

Why does anything to the power zero equal one?

Because dividing a power by itself gives 1, and by the exponent rule it also gives that base to the power zero. Both must agree, so any nonzero base to the zero is 1.

What is zero to the power zero?

It is a special case with no single agreed answer. Many contexts define 0^0 as 1 for convenience, but strictly it is treated as indeterminate. Most everyday exponent rules assume a nonzero base.

How is 2^3 different from 2 × 3?

2 × 3 adds 2 to itself three times and equals 6. 2^3 multiplies 2 by itself three times and equals 8. Exponents grow much faster than multiplication.

Can an exponent be a decimal?

Yes. A decimal exponent is just a fractional one written differently: 2^0.5 is the same as 2^(1/2), the square root of 2, about 1.414. Calculators handle any real exponent this way.