The value $$\left(\frac{1+\sqrt 3}{2\sqrt 2}+\frac{\sqrt 3-1}{2\sqrt 2}i\right)^{72}$$ is a positive real number. What real number is it?

We can write

[   (1 + sqrt (3)   +  sqrt (3)i  - 1i  ] ^72  / sqrt (8)^72  =

[  ( 1 - i)   +  (1 + i) sqrt(3) ] ^72  / sqrt (8) ^72  =

[ [ (1 - i)  + (1 + i)sqrt(3) ]^2 ] ^36 / [sqrt (8)^2]^36 =

[ [ (1 - i)  + (1 + i)sqrt(3)]^2 / sqrt (8)^2 ] ^36

Forget  about  the exponent of 36  for a second and look at this

[ (1 - i)   +  (1 + i) sqrt (3) ] ^2 / [sqrt (8)]^2  =

[ (1 - i)(1 - i)   +  2 (1 - i) (1 + i)sqrt (3)  +  (1 + i)(1 + i) *3 ]/8   =

[  - 2i  + 2 ( 2)sqrt (3)  + 2i * 3 ]/ 8    =

[ 4i + 4 sqrt (3) ]  =   [  4  (  i  +  sqrt (3) ) ]  / 8   =

(1/2)  [ sqrt (3)  + i ]

So.......we have

[ (1/2)^36]  *  [ sqrt (3)  + i ]^36    =   A

Looking at  the red part we can write this as

(2)^36  ( cos [36 (pi/6)]  + i sin [ 36 (pi/6) ] )  =

(2)^36  (  cos [ 6pi ]  +  i sin [6pi] )  =

(2)^36  [  1  +  0i ]  =

(2)^36

So  A   becomes

[ (1/2)^36 ] * 2^36  =  [ 2 / 2 ] ^36  =  1^36  =

1

((1/2 + i/2) (sqrt(3) - i))/sqrt(2)=
Divide: 1 / 2 = 0.5
Divide: i / 2 = 0.5i
Square root: sqrt(3) = √ 3 = 1.7320508
Multiple: (0.5+0.5i) * (1.7320508-i) = 0.866025403784-0.5i+0.866025403784i+0.5 = 1.3660254+0.3660254i
alternative steps
0.7071068 × ei 45° × 2 × ei (-30°) = 0.707106781187 × 2 × ei (45°+(-30°)) = 1.4142136 × ei 15° = 1.3660254+0.3660254i
Square root: sqrt(2) = √ 2 = 1.4142136
Divide: (1.3660254+0.3660254i) / 1.4142136 = 0.9659258+0.258819i =(-1)^(1/12)
(-1)^(1/12)*72 =(-1)^6 =1