show that the following is an integer

Question

(√2)∧(√2) ∧(√2) ∧(√2) ∧.......

Answer 1

    let    x=(√2)∧(√2) ∧(√2) ∧(√2) ∧.......

then we can also write  

 x= (√2)^x   ;  (as x is made up of infinite no. of terms so there will be no effect in the no. of terms if we write like this)

 

                             So  x^1/x= ( (√2)^x )^(1/x)

                                  x^1/x   = √2                                As √2 ^(x*(1/x))= √2

  or we can write as       x^1/x = 2^1/2

 

comparing both sides we will get x= 2

So        x =   (√2)∧(√2) ∧(√2) ∧(√2) ∧....... = 2