Question 1152514
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Consider this calculation:<br>
{{{(sqrt(a)+sqrt(b))^2}}}<br>
= {{{(a+b)+2sqrt(ab)}}}<br>
So if you have an expression in the form {{{(a+b)+2sqrt(ab)}}}, then its square root is {{{sqrt(a)+sqrt(b)}}}.<br>
To find the square root of the given expression, you need to put it in exactly that form -- specifically, you need the radical part to be {{{2sqrt(ab)}}}.  So<br>
{{{sqrt(32-6sqrt(15)) = sqrt(32-2*(3sqrt(15))) = sqrt(32-2sqrt(135))}}}<br>
And then the task is to find the numbers a and b for which a+b=32 and ab=135.<br>
Those numbers are 27 and 5, so<br>
{{{sqrt(32-6sqrt(15)) = sqrt(27)-sqrt(5) = 3sqrt(3)-sqrt(5)}}}<br>