<PRE><B>Question 1127390</B>
&lt;br&gt;
One general method for evaluating expressions like this is to consider an expression of the form&lt;br&gt;
{{{(sqrt(a)+sqrt(b))^2}}}&lt;br&gt;
Expanded, this expression is equal to&lt;br&gt;
{{{(a+b)+2sqrt(ab)}}}&lt;br&gt;
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.&lt;br&gt;
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.&lt;br&gt;
{{{sqrt(7/12)-sqrt(3)/3 = sqrt(21/36)-12*sqrt(3)/36}}}&lt;br&gt;
Then&lt;br&gt;
{{{sqrt(sqrt(21/36)-12*sqrt(3)/36) = (sqrt((21)-12*sqrt(3)))/6}}}&lt;br&gt;
Now we need to simplify&lt;br&gt;
{{{(sqrt((21)-12*sqrt(3)))}}}&lt;br&gt;
using the process described at the beginning of my response.&lt;br&gt;
We need to rewrite the expression under the radical in the exact required form, with a coefficient of 2 on the inside radical:&lt;br&gt;
{{{(sqrt((21)-2*sqrt(108)))}}}  [divide outside the radical by 6; multiply inside by sqrt(36)]&lt;br&gt;
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&lt;br&gt;
{{{(sqrt((21)-2*sqrt(108))) = sqrt(12)-sqrt(9)}}}&lt;br&gt;
Then to finish the problem we divide by 6:&lt;br&gt;
{{{sqrt(12)/6-sqrt(9)/6 = 2*sqrt(3)/6-3/6 = sqrt(3)/3-1/2}}}&lt;br&gt;
The directions say to express the solution in the form x + sqrt(y).  So we have to express sqrt(3)/3 as sqrt(y):&lt;br&gt;
{{{sqrt(3)/3 = sqrt(3)/sqrt(9) = sqrt(3/9) = sqrt(1/3)}}}&lt;br&gt;
So...&lt;br&gt;
ANSWER: {{{sqrt(sqrt(7/12)-sqrt(3)/3) = (-1/2)+sqrt(1/3)}}}</PRE>