Laws of Indices

When working with indices, there are three rules that should be used:

When m and n is positive integers,

1.	am × an = am + n
2.	am ÷ an = am – n   or	am an	= am – n   (m ≥ n)
3.	(am)n = am × n

This is the definition of aⁿ logical consequences, these three are the results, but formal proof are really needed.With particular examples as below, you can 'verify' them, but this are not the proof.

27× 23	=	(2 × 2 × 2 × 2 × 2 × 2 × 2) × (2 × 2 × 2)
	=	2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
	=	210	(here m = 7, n = 3 and m + n = 10)

or,

27 ÷ 23	=	2 × 2 × 2 × 2 × 2 × 2 × 2 2 × 2 × 2
	=	2 × 2 × 2 × 2
	=	24	(again m = 7, n = 3 and m – n = 4)

Also,

(27)3	=	27 × 27 × 27
	=	221	(using rule 1)   (again m = 7, n = 3 and m × n = 21)

Below, the first rule proof is given below:

Proof

am × an	=	a × a × ... × a m of these	×	a × a × ... × a n of these
	=	a × a × ... × a × a × a × ... × a (m+n) of these
	=	am+n

In a similar way, for all positive integers m and n, the second and third rules can be shown.
Using rule 2 we can see an important result:

	xn xn	= xn – n = x0
but	xn xn	= 1,	so

x⁰ = 1

This is true for any non-zero value of x, so, for example, 3⁰ = 1, 27⁰ = 1 and 1001⁰ = 1.