Question

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

Answer 1

See explanation...

Explanation:

"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`"