Question

State the quotient law of exponents.

Answer 1

The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. If a is a non-zero real number and m, n are positive integers, then  `a^m/a^n = a^(m-n)`

We shall divide the proof into three parts 

(i) when  m>n

(ii) when  m  = n 

(iii) when  m < n

Case 1 

When   m > n

We have 

\[\frac{a^m}{a^n} = \frac{a \times a \times a . . . .\text {  to m factors }}{a \times a \times a . . . . \text { to n factors }}\]

\[\frac{a^m}{a^n} = a \times a \times a . . . . to (m - n) \text { factors }\]

\[\frac{a^m}{a^n} = a^{m - n}\]

Case 2 

When  m = n

We get

 `a^m/a^n = a^m/a^m`

Cancelling common factors in numerator and denominator we get,

`a^m/a^n = 1`

By definition we can write 1 as a°

 `a^m/a^n = a^(m-m)`

 `a^m/a^n = a^(m-n)`

Case 3 

When  m < n

In this case, we have 

`a^m/a^n = 1/(axx axx a ....(n-m))`

`a^m/a^n = 1/(a^(n-m))`

`a^m/a^n = a^-(n-m)`

`a^m/a^n = a^(m-n)`

Hence `a^m/a^n = a^(m-n)`, whether m < n, m = n or,m > n