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

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Answer 1 · 2016-09-02T19:24:33

#sqrt(1/3), sqrt(1/2), sqrt(2/3), sqrt(3/4)#

Given:

#sqrt(1/2), sqrt(1/3), sqrt(2/3), sqrt(3/4)#

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

#1/2, 1/3, 2/3, 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#)...

#{(1/2 = 6/12), (1/3 = 4/12), (2/3 = 8/12), (3/4 = 9/12) :}#

Hence the correct order of the radicands is:

#1/3, 1/2, 2/3, 3/4#

and the correct order of the square roots is:

#sqrt(1/3), sqrt(1/2), sqrt(2/3), sqrt(3/4)#