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

Jun 17, 2017

See explanation...

#### Explanation:

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

• $\sqrt{\frac{3}{2}} \text{ }$ "the square root of: $3$ divided by $2$"

• $\frac{\sqrt{3}}{2} \text{ }$ "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 $n \ge 0$ 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...

$\sqrt{\frac{3}{2}} = \sqrt{\frac{6}{4}} = \sqrt{\frac{6}{{2}^{2}}} = \frac{\sqrt{6}}{\sqrt{{2}^{2}}} = \frac{\sqrt{6}}{2}$

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 $2$"