Question 1189763
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Consider the general expansion<br>
{{{(a+sqrt(b))^2=(a^2+b)+2a(sqrt(b))}}}<br>
You can use that pattern to attempt to find the value of an expression of the form<br>
{{{sqrt(m+sqrt(n))}}}<br>
In this problem, the expression under the radical on the left is<br>
{{{2*sqrt(10)+11}}} or {{{11+2*sqrt(10)}}}<br>
According to the pattern above, we should have {{{a^2+b=11}}}, {{{a=1}}}, and {{{b=10}}}.<br>
And those are all satisfied when a=1 and b=10.  So<br>
{{{sqrt(11+2*sqrt(10)) = 1+sqrt(10)}}}<br>
Replacing the left side of the given equation with that gives us<br>
{{{1+sqrt(10)=x+sqrt(x+y)}}}<br>
Solving by equating the rational and irrational parts of the two expressions gives us x=1 and y=9; so then<br>
ANSWER: x-y = -8<br>