Question

Can you find two integers m,n such that $2^{m+n} =2^{mn}$?

Answer 1

Solution :

************************************
We know the Exponential Law :

If a^m = a^n then m = n

************************************

Here ,

2^(m+n) = 2^mn

=> m + n = mn

=> n = mn - m

=> n = m( n - 1 )

=> n/( n - 1 ) = m

=> m = n/( n - 1 )

*******************************
But n/( n - 1 ) is an integer .

When n - 1 = 1

=> n = 1 + 1 = 2
********************************

=> m = 2/( 2 - 1 )

=> m = 2

Therefore ,

If m = n = 2,

then

2^(m+n) = 2^(mn)

••••••