# How do you order the following from least to greatest sqrt(1/2), sqrt(1/3), sqrt(2/3), sqrt(3/4)?

Sep 2, 2016

$\sqrt{\frac{1}{3}} , \sqrt{\frac{1}{2}} , \sqrt{\frac{2}{3}} , \sqrt{\frac{3}{4}}$

#### Explanation:

Given:

$\sqrt{\frac{1}{2}} , \sqrt{\frac{1}{3}} , \sqrt{\frac{2}{3}} , \sqrt{\frac{3}{4}}$

Note that square roots increase monotonically with the radicand, so all we need to do is order the radicands:

$\frac{1}{2} , \frac{1}{3} , \frac{2}{3} , \frac{3}{4}$

One way of making that easier is to give them all a common denominator $12$ (being the least common multiple of $2 , 3 , 4$)...

$\left\{\begin{matrix}\frac{1}{2} = \frac{6}{12} \\ \frac{1}{3} = \frac{4}{12} \\ \frac{2}{3} = \frac{8}{12} \\ \frac{3}{4} = \frac{9}{12}\end{matrix}\right.$

Hence the correct order of the radicands is:

$\frac{1}{3} , \frac{1}{2} , \frac{2}{3} , \frac{3}{4}$

and the correct order of the square roots is:

$\sqrt{\frac{1}{3}} , \sqrt{\frac{1}{2}} , \sqrt{\frac{2}{3}} , \sqrt{\frac{3}{4}}$