sqrt(3/2)*sqrt(1/2) in surd form?

Simplify the following:
sqrt(3/2) sqrt(1/2)

sqrt(1/2)  =  (sqrt(1))/(sqrt(2)):
sqrt(3/2) (sqrt(1))/(sqrt(2))

Rationalize the denominator. (sqrt(1))/(sqrt(2))  =  (sqrt(1))/(sqrt(2))×(sqrt(2))/(sqrt(2))  =  (sqrt(1) sqrt(2))/(2):
sqrt(3/2) (sqrt(1) sqrt(2))/(2)

sqrt(1) sqrt(2) = sqrt(1×2):
(sqrt(3/2) sqrt(2))/(2)

sqrt(3/2)  =  (sqrt(3))/(sqrt(2)):
((sqrt(3))/(sqrt(2)) sqrt(2))/(2)

Rationalize the denominator. (sqrt(3))/(sqrt(2))  =  (sqrt(3))/(sqrt(2))×(sqrt(2))/(sqrt(2))  =  (sqrt(3) sqrt(2))/(2):
((sqrt(3) sqrt(2))/(2) sqrt(2))/(2)

sqrt(3) sqrt(2) = sqrt(3×2):
(sqrt(3×2) sqrt(2))/(2×2)

3×2  =  6:
( sqrt(6 )  sqrt(2))/(2×2)

sqrt(6) sqrt(2) = sqrt(6×2):
(sqrt(6×2))/(2×2)

6×2  =  12:
( sqrt(12 ) )/(2×2)

sqrt(12)  =  sqrt(2^2×3)  =  2 sqrt(3):
(2 sqrt(3))/(2×2)

(2 sqrt(3))/(2×2) = (sqrt(3))/(2):
Answer: |  (sqrt(3))/(2)

Another way to get the answer:

Knowing that sqrt(A) * sqrt(B)  =  sqrt(A*B),

sqrt(3/2) * sqrt(1/2)  =  sqrt( 3*1 / 2*2 )  =  sqrt(3/4).

Knowing that sqrt(A/B)  =  sqrt(A) / sqrt(B),

--->   sqrt(3/4)  =  sqrt(3) / sqrt(4)  =  sqrt(3) / 2.

thankyou！