Question

What is the square root of $3$ divided by $2$ ?

Answer 1

"the square root of $3$ divided by $2$" could mean either of the following:
$\sqrt{\frac{3}{2}}$ "the square root of $3$ divided by $2$"
$\frac{\sqrt{3}}{2}$ "the square root of $3,$ divided by $2$".
A square root of a number n is a number x, such that $x^2=n.$ Every non-zero number actually has two square roots, which we call $\sqrt{n}$ and $-\sqrt{n}$ When we say "the" square root, we usually mean the principal one $\sqrt{n},$ which $n\ge 0$ is the non-negative one.
In either of the above interpretations of the question, the resulting number will be an irrational number - not a rational one.
Considering each in turn:
We can "simplify" the first square root
Note that if $a,b>0$ then $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}},$ so...
$\sqrt{\frac{3}{2}}=\sqrt{\frac{6}{4}}n=\sqrt{\frac{6}{2^2}}=\frac{\sqrt{6}}{\sqrt{2^2}}=\frac{\sqrt{6}}{2}$
We have:
$\sqrt{\frac{3}{2}}=\frac{\sqrt{6}}{2}\approx 1.2247$
The second expression cannot be simplified in that way:
$\frac{\sqrt{3}}{2}$
is in simplest form.
As an approximation, we can write:
$\frac{\sqrt{3}}{2}\approx 0.8660$
This particular number is important as it occurs as the height of an equilateral triangle with sides of length 1. More commonly, to separate out the divisor 2, we consider an equilateral triangle of side 2 and bisect it...
Hence we find that:
$sin\left(\frac{\pi }{3}\right)=cos\left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}$
So when you encountered the expression "the square root of 3 divided by 2" it seems likely to me that the intention was:
"the square root of $3,$ divided by $2$"