Zero and Negative Indices

Step the index down

Write the powers of 2 from the top down: $2^{3} = 8$, $2^{2} = 4$, $2^{1} = 2$. Each step down lowers the index by 1 and takes away one copy of 2, so the value is divided by 2: 8, then 4, then 2.

The zero index

Take one more step down. The index goes from 1 to 0, and the value is divided by 2 once more: 2 ÷ 2 = 1. So $2^{0} = 1$. Any other value would break the pattern.

The same happens for every base. The powers of 3 run 27, 9, 3, each a third of the one before, and one more step gives 3 ÷ 3 = 1, so $3^{0} = 1$. Any base other than 0, raised to the power 0, is 1.

Each step down the index halves the bar: 8, 4, 2, and then 1.

The dividing law agrees

The law for dividing powers gives the same answer. $2^{3} / 2^{3} = 2^{3-3} = 2^{0}$, and dividing 8 by 8 gives 1. Every copy cancels, and what is left is 1, not 0.

All three copies cancel. The value is 8 ÷ 8 = 1.

Below zero

Keep going. One step below $2^{0}$ divides 1 by 2: $2^{-1} = \frac{1}{2}$. The next step gives $2^{-2} = \frac{1}{4}$, and the next $2^{-3} = \frac{1}{8}$.

Compare these with the powers above zero. $\frac{1}{2}$ is 1 divided by $2^{1}$, $\frac{1}{4}$ is 1 divided by $2^{2}$, and $\frac{1}{8}$ is 1 divided by $2^{3}$. A negative index means one over the power: $2^{-3}$ is 1 divided by $2^{3}$, which is $\frac{1}{8}$. In the same way, $5^{-2}$ is 1 divided by $5^{2}$, which is $\frac{1}{25}$.

The dividing law agrees again: $2^{2} / 2^{3} = 2^{2-3} = 2^{-1}$, and $4 \div 8 = \frac{1}{2}$.

Past $2^{0}$ the halving carries on: $\frac{1}{2}$, then $\frac{1}{4}$. The values get smaller, but they never reach 0 or go below it.

The usual mistakes

A zero index does not give 0. $5^{0} = 1$, because each step down divides by 5, and 5 ÷ 5 = 1.

A negative index does not make the value negative. $2^{-3}$ is $\frac{1}{8}$, a small positive number, not −8.

$2^{-3}$ is not $\frac{1}{6}$ either. The index counts copies, so $2^{3} = 8$, and $2^{-3}$ is $\frac{1}{8}$.