Question 81

$$\sqrt{30 - 12\sqrt{6}} =$$


Different approaches are possible for this question. 
One of the ways can be to take the square of any option and it should result in the term which is enclosed in the square root sign in the question.

Another approach would be to split the terms of the square root to make it a perfect square.
$$30-12\sqrt{\ 6}=30-2\cdot6\cdot\sqrt{\ 3}\sqrt{\ 2}=30-2\cdot3\cdot2\cdot\sqrt{\ 3}\sqrt{\ 2}$$

It means 30 could be the sum of squares of one of the following possible combinations: $$3\sqrt{\ 3\ }\&\ 2\sqrt{\ 2}$$ OR $$3\sqrt{\ 2\ }\&\ 2\sqrt{\ 3}$$
We can see that it is the sum of the squares of the second combination. 

Hence, $$30-12\sqrt{\ 6}$$ = $$\left(3\sqrt{\ 2\ }-2\sqrt{\ 3}\right)^2$$

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