Mathematics

Problem

Let

A = 8^{\sqrt{2}}, \quad B = 4^{\sqrt{3}}, \quad C = 2^{3\sqrt{2}}

. Answer the following questions: (i) Express

A, B

and

C

as powers of 2, writing each in the form

2^{k}

where

k

is a positive real number. (ii) Arrange

A, B

and

C

in descending order of magnitude. (iii) Provide full justification for the order obtained, citing the comparisons of their exponents.