Second index law

Example

Consider

Example

More generally, if m > n,

2m = 2m n

2n

Fact

Even more generally, if m > n,

xm = xm n

xn

Third index law

Example

Example

More generally,

(2m )n = 2mn

Fact

Even more generally,

(xm )n = xmn

Fourth index law

In maths textbooks, you may have seen an index or power 0, such as x0. What can this possibly mean?

Interestingly enough, we can attach a meaning to a zero power by requiring it to obey the laws already established.

Example

If expressions involving the zero power are to obey the …rst law, then, for example, 23 20 = 23+0 = 23. So we have 23 20 = 23.

Example

More generally, xn x0 = xn+0 = xn. This can only mean x0 = 1.

Fact

Since xn x0 = xn, the only reasonable de…nition for x0 is that x0 = 1.

Fifth index law

We may extend these ideas still further to power 1, such as 2 1.

Again, we require it to obey the laws already established. We may then attach a meaning to power 1 as follows.

Example

If expressions involving the power 1 are to obey the …rst and fourth laws,

then, for example, 21 2 1 = 21+( 1) = 20 = 1. So we have 21 2 1 = 1.

Example

More generally, x1 x 1 = x0 = 1

Fact

Since x1 x 1 = 1, the only reasonable de…nition for x 1 is that x 1 = 1/x.