Question

find all ordered pairs of positive integers (x,y) that satisfy the following equation: x^y=2^64. please answer and explain how you got it!!!!

Answer 1

2^64 can be written as a power in the following ways

2^64 , 4^32 , 16^16, 256^8 , 65536^4, 4294967296^2 and (4294967296^2 )^1

2^64 = 2x2x2x2...x2x2 --- 64 of them

(2x2)x(2x2)x2...x2)x(2x2) --- 32 pairs of 2's
= 4^32

(2x2x2x2)x(2x2...x2x(2x2x2x2) --- 16 groups of 4 2's
= 16^16
etc

e.g. I can only split the 64 into equal groups,
I could not use three 2's, since 64 does not divide evenly by 3

so ordered pairs of whole positive numbers are
(2,64) , (4,32), (16,16) , (256,8) , (655336, 4)
(4294967296,2) and (4294967296^2 , 1)