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

"the square root of #3# divided by #2#" could mean either of the following:

• #sqrt(3/2)" "# "the square root of: #3# divided by #2#"

• #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 for #n >= 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(a/b) = sqrt(a)/sqrt(b)#, so...

#sqrt(3/2) = sqrt(6/4) = sqrt(6/(2^2)) = sqrt(6)/sqrt(2^2) = sqrt(6)/2#

We have:

#sqrt(3/2) = sqrt(6)/2 ~~ 1.2247#

The second expression cannot be simplified in that way:

#sqrt(3)/2#

is in simplest form.

As an approximation, we can write:

#sqrt(3)/2 ~~ 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(pi/3) = cos(pi/6) = 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#"