Question 1209471
<br>
Consider the general expression<br>
{{{(sqrt(a)+sqrt(b))^2}}}<br>
Expanding this, we get<br>
{{{a+2sqrt(ab)+b}}} or {{{(a+b)+2sqrt(ab)}}}<br>
Turning this around, we have the general result<br>
[*] {{{sqrt((a+b)+2sqrt(ab))=sqrt(a)+sqrt(b)}}}<br>
In this problem, we have the expression<br>
{{{sqrt(11+2sqrt(10))}}}<br>
Comparing that to the general result[*], we can see by inspection that<br>
{{{sqrt(11+2sqrt(10))=sqrt(1)+sqrt(10)=1+sqrt(10)}}}<br>
The problem asks us to write that in the form<br>
{{{x+sqrt(x+y)}}}<br>
So we want to have<br>
x = 1  and x+y = 10<br>
which leads us immediately to<br>
x = 1 and y = 9.<br>
Finally, the question asks for the value of x-y.<br>
ANSWER: x-y = 1-9 = -8<br>