Question 1030250
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Consider this patern:<br>
{{{(sqrt(a)+sqrt(b))^2=(a+b)+2sqrt(ab)}}}<br>
Given an expression to be simplified in the form {{{sqrt(a+b*sqrt(c))}}}, if you can modify the expression so the coefficient "b" on the inside radical is 2, then the expression can be simplified using that pattern.<br>
In your example, you can take "4" out of the coefficient "8", leaving the required coefficient "2"; the "4" goes back inside the radical as {{{sqrt(16)}}}:<br>
{{{sqrt(24+8sqrt(5))=sqrt(24+2sqrt(16*5))=sqrt(24+2sqrt(80))}}}<br>
The pattern now says you want to find integers a and b for which a+b=24 and ab=80.  Those integers are 20 and 4, so<br>
{{{sqrt(24+8sqrt(5))=sqrt(24+2sqrt(80))=sqrt(20)+sqrt(4)=2sqrt(5)+2}}}<br>