SOLUTION: Let {{{ sqrt(7/12 - sqrt(3)/3 ) = x+sqrt(y)}}} , where x<0 and y>0. By solving for x and y, simplify {{{ sqrt(7/12

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Question 1127390: Let , where x<0 and y>0. By solving for x and y, simplify .
Answer by greenestamps(13198) (Show Source):
You can put this solution on YOUR website!

One general method for evaluating expressions like this is to consider an expression of the form

Expanded, this expression is equal to

Then solving for the square root of that expression means finding two numbers a and b whose sum is the real part of the expression and whose product is the radicand of the inner square root.

Using that process directly, with the fractions, will be a bit awkward; to simplify the process, put the expression in the radical in a form with a common denominator that is a perfect square. That will allow us to take out the fractions using a common denominator, leaving us integers to work with.

Then

Now we need to simplify

using the process described at the beginning of my response.

We need to rewrite the expression under the radical in the exact required form, with a coefficient of 2 on the inside radical:

[divide outside the radical by 6; multiply inside by sqrt(36)]

Now, using the form shown earlier, we look for two numbers with a sum of 21 and a product of 108; the numbers are 9 and 12. So

Then to finish the problem we divide by 6:

The directions say to express the solution in the form x + sqrt(y). So we have to express sqrt(3)/3 as sqrt(y):

So...

ANSWER:

SOLUTION: Let {{{ sqrt(7/12 - sqrt(3)/3 ) = x+sqrt(y)}}} , where x<0 and y>0. By solving for x and y, simplify {{{ sqrt(7/12 - sqrt(3)/3 )}}}.