prove, using the law of exponents, that the following is true

Question

(2^4)^3=(64)^2
(use the same bases to rewrite the expression)
the second half of =(8)^2 (8)^2
=(4)^2 (2)^2 (4)^2 (2)^2
=(2)^2(2^2)(2^2)(2^2)(2^2)(2^2)
now the first part
(2^4)^3
2x2x2x2
2x2x2x2
2x2x2x2
so put both sides together and get
2x2x2x2=2x2x2x2
2x2x2x2=2x2x2x2
2x2x2x2=2x2x2x2
Is this correct?
Thanks for checking my work

Answer 1

( 2 ^ 4 ) ^ 3 = ( 2 ^ 3 ) ^ 4 = 8 ^ 4

64 ^ 2 = ( 8 ^ 2 ) ^ 2 = 8 ^ 4

Answer 2

but the question said it the expression should have the same base, so wouldn't you have to go further?

Answer 3

changing to the same base

(2^4^3
= 2^12

64^2
= (2^6)^2 = 2^12

therefore (2^4)^3 = 64^2

Answer 4

I thought having the same base meant having the same number if it can be in simplier form. that is why I made them all multiples of 2.
Thank you for your help