Question 1107093
<pre>
One pair of integers (x.y) solves {{{ sqrt( 4*sqrt( 7 )+11 )=y+sqrt( x ) }}}, find x*y.
***************************************************************<font face = tahoma><font size = 2><font color = blue><b>
                              {{{sqrt(4sqrt(7) + 11))}}}
                              {{{sqrt(11 + 4sqrt(7))}}}
                    {{{sqrt(11 + 2(2)sqrt(7))}}} ----- Replacing 4 with its PRIME factors, 2 & 2
                         {{{sqrt(11 + 2sqrt(4)sqrt(7))}}} ----- Converting 2 to {{{sqrt(4)}}}                 
                      {{{sqrt(7 + 4 + 2sqrt(7)sqrt(4))}}} ---- Changing 11 to 7 + 4             
     {{{sqrt((sqrt(7))^2 + (sqrt(4))^2 + 2sqrt(7)sqrt(4))}}} ---- Converting {{{system(matrix(2,3, 7, to, (sqrt(7))^2, 4, to, (sqrt(4))^2))}}}
The above is in the form: {{{(a + b)^2}}}, with {{{system(matrix(2,3, a, being, sqrt(7), b, being, sqrt(4)))}}}, and so:
{{{sqrt((sqrt(7))^2 + (sqrt(4))^2 + 2sqrt(7)sqrt(4))}}} then becomes: {{{sqrt((sqrt(7) + sqrt(4))^2)}}} 
                                                                                {{{sqrt(7) + sqrt(4)}}} ----- Cancelling SQUARE and SQUARE ROOT
                                                                                {{{sqrt(7) + 2}}}

√(4√(7+11)) = {{{sqrt(7) + 2}}}, is in the form: y + √x, with y = 2, and x = 7. Therefore, x*y = 7(2) = 14. </font></font></font></b></pre>