How do you order the following from least to greatest $\sqrt{\frac{1}{2}}, \sqrt{\frac{1}{3}}, \sqrt{\frac{2}{3}}, \sqrt{\frac{3}{4}}$ $sqrt(1/2), sqrt(1/3), sqrt(2/3), sqrt(3/4)$ ?

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Answer 1 · 2016-09-02T19:24:33

$\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{2}}, \sqrt{\frac{2}{3}}, \sqrt{\frac{3}{4}}$ $sqrt(1/3), sqrt(1/2), sqrt(2/3), sqrt(3/4)$

Given:

$\sqrt{\frac{1}{2}}, \sqrt{\frac{1}{3}}, \sqrt{\frac{2}{3}}, \sqrt{\frac{3}{4}}$ $sqrt(1/2), sqrt(1/3), sqrt(2/3), sqrt(3/4)$

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

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

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

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

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:

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

How do you order the following from least to greatest sqrt(1/2), sqrt(1/3), sqrt(2/3), sqrt(3/4)√12,√13,√23,√34sqrt(1/2), sqrt(1/3), sqrt(2/3), sqrt(3/4)?