Question 1209214
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Answer: <font color=red>444</font>


Explanation
I'll assume that a,b are positive integers.


{{{sqrt(80+sqrt(1776))=sqrt(a) +sqrt(b)}}}


{{{(sqrt(80+sqrt(1776)))^2=(sqrt(a) +sqrt(b))^2}}} Square both sides


{{{80+sqrt(1776)=(sqrt(a))^2+2*sqrt(a)*sqrt(b)+(sqrt(b))^2}}}


{{{80+sqrt(1776)=a+2*sqrt(ab)+b}}} 


{{{80+sqrt(1776)=(a+b)+2*sqrt(ab)}}}


{{{highlight(80)+highlight_green(sqrt(1776))=highlight(a+b)+highlight_green(2*sqrt(ab))}}} Notice these matching pairs on either side


The red and green boxes indicate that {{{a+b = 80}}} and {{{2*sqrt(ab) = sqrt(1776)}}}


Rewrite the 2nd equation like so
{{{2*sqrt(ab) = sqrt(1776)}}}


{{{(2*sqrt(ab))^2 = (sqrt(1776))^2}}}


{{{4ab = 1776}}}


{{{ab = 1776/4}}}


{{{ab = 444}}}



Extra info:
Solving the system {{{system(a+b = 80, 2*sqrt(ab) = sqrt(1776))}}} leads to (a,b) = (6,74) when considering a < b 
a+b = 6+74 = 80
a*b = 6*74 = <font color=red>444</font>
{{{sqrt(80+sqrt(1776))=sqrt(6) +sqrt(74)}}}
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