How do I simplify 2sqrt(6)/3 into 2sqrt(2/3)?

[souljagurl]

How do I simplify 2sqrt(6)/3 into 2sqrt(2/3)? I am in calc 3 and still having trouble with simple reducing fractions problems

 

[souljagurl]

Thanks a ton!

 

[soroban]

Hello, souljagurl!

\(\displaystyle \text{How do I simplify }\dfrac{2\sqrt{6}}{3}\,\text{ into }\,2\sqrt{\dfrac{2}{3}}\,?\)

This is an unusual problem; it is the opposite of simplifying.
Usually, we do not want radicals in the denominator
. . nor denominators under a radical.

We have: \(\displaystyle \;\dfrac{2\sqrt{6}}{3} \;=\;\dfrac{2\cdot\sqrt{2}\cdot\sqrt{3}}{3}\)

Multiply by \(\displaystyle \frac{\sqrt{3}}{\sqrt{3}}\!:\;\;\dfrac{2\cdot\sqrt{2}\cdot\sqrt{3}}{3}\cdot\dfrac{\sqrt{3}}{\sqrt{3}} \;=\;\dfrac{2\cdot\sqrt{2}\cdot3}{3\sqrt{3}}\). .

. . . . . . . . \(\displaystyle =\;\dfrac{2\sqrt{2}}{\sqrt{3}} \;=\;2\sqrt{\dfrac{2}{3}} \)