Question 824392
<pre>
I need detail solution:
A = ( {{{sqrt(2)}}} + {{{sqrt(3)}}} + {{{sqrt(5)}}})( {{{sqrt(2)}}} &#8722; {{{sqrt(3)}}} + {{{sqrt(5)}}}) and B = ( {{{sqrt(2)}}} + {{{sqrt(3)}}} &#8722; {{{sqrt(5)}}})(&#8722;{{{sqrt (2)}}} + {{{sqrt(3)}}} + {{{sqrt(5)}}}) find C=AB

(In the "A" I've got 4-2{{{sqrt(15)}}}, but I can't solve B)
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A = {{{system((sqrt(2) + sqrt(3) - sqrt(5))(sqrt(2) - sqrt(3) + sqrt(5)), 
  (sqrt(2) + sqrt(5) + sqrt(3))(sqrt(2) + sqrt(5) - sqrt(3)),  
  ((sqrt(2) + sqrt(5)) + sqrt(3))((sqrt(2) + sqrt(5)) - sqrt(3)), 
  (sqrt(2) + sqrt(5))^2 - (sqrt(3))^2, 
  (sqrt(2))^2 + 2sqrt(2)sqrt(5) + (sqrt(5))^2 - 3,
    2 + 2sqrt(10) + 5  -  3,
    2 + 5 - 3 + 2sqrt(10),
    highlight(4 + 2sqrt(10)))}}}           B = {{{system( (sqrt(2) + sqrt(3) - sqrt(5))(- sqrt(2) + sqrt(3) + sqrt(5)), 
  (sqrt(3) - sqrt(5) + sqrt(2))(sqrt(3) + sqrt(5) - sqrt(2)),  
  ((sqrt(3) - sqrt(5)) + sqrt(2))((sqrt(3) + sqrt(5)) - sqrt(2)), 
   (sqrt(3) - sqrt(5))( sqrt(3) + sqrt(5)) - sqrt(2)(sqrt(3) - sqrt(5)) + sqrt(2)(sqrt(3) + sqrt(5)) - (sqrt(2))^2,
(sqrt(3))^2 - (sqrt(5))^2 - sqrt(2)sqrt(3) + sqrt(2)sqrt(5) + sqrt(2)sqrt(3) + sqrt(2)sqrt(5) - 2,
    3 - 5 - sqrt(6) + sqrt(10) + sqrt(6) + sqrt(10)  -  2,
    3 - 5 - 2 + 2sqrt(10),
    highlight(- 4 + 2sqrt(10))))}}}

C = {{{system(AB, 
(4 + 2sqrt(10))(- 4 + 2sqrt(10)), - 16 + 8sqrt(10) - 8sqrt(10) + (2sqrt(10))^2,  - 16 + 4(10), matrix(1,3, - 16 + 40, "=", highlight(24)))}}}

                              OR

A = ({{{sqrt(2)}}} + {{{sqrt(3)}}} + {{{sqrt(5)}}})( {{{sqrt(2)}}} - {{{sqrt(3)}}} + {{{sqrt(5)}}}) 
A = {{{sqrt(2)(sqrt(2) - sqrt(3) + sqrt(5)) + sqrt(3)(sqrt(2) - sqrt(3) + sqrt(5)) + sqrt(5)(sqrt(2) - sqrt(3)  + sqrt(5))}}} 
A = {{{sqrt(2)^2 - sqrt(2)sqrt(3) + sqrt(2)sqrt(5)}}} + {{{sqrt(3)sqrt(2) - (sqrt(3))^2 + sqrt(3)sqrt(5)}}} + {{{sqrt(5)sqrt(2) - sqrt(5)sqrt(3)  + (sqrt(5))^2}}}
A = {{{2 - sqrt(6) + sqrt(10)}}} + {{{sqrt(6) - 3 + sqrt(15)}}} + {{{sqrt(10) - sqrt(15) + 5}}}
A = {{{2 - 3 + 5 - sqrt(6) + sqrt(6) + sqrt(10) + sqrt(10) + sqrt(15) - sqrt(15)}}} ----- COLLECTING like-terms
A = {{{4 + sqrt(10) + sqrt(10)}}} = {{{highlight(4 + 2sqrt(10))}}} <=== CORRECT value of A, so yours is INCORRECT

B = {{{system((sqrt(2) + sqrt(3) - sqrt(5))(- sqrt(2) + sqrt(3) + sqrt(5)), sqrt(2)(- sqrt(2) + sqrt(3) + sqrt(5)) + sqrt(3)(- sqrt(2) + sqrt(3) + sqrt(5)) - sqrt(5)(- sqrt(2) + sqrt(3) + sqrt(5)), - (sqrt(2))^2 + sqrt(2)sqrt(3) + sqrt(2)sqrt(5) + sqrt(3)(- sqrt(2)) + (sqrt(3))^2 + sqrt(3)sqrt(5) - (sqrt(5))(- sqrt(2)) - sqrt(5)sqrt(3) - (sqrt(5))^2, - 2 + sqrt(6) + sqrt(10) - sqrt(6) + 3 + sqrt(15) + sqrt(10) - sqrt(15) - 5, matrix(1,4, - 2 + 3 - 5 + sqrt(6) - sqrt(6) + sqrt(10) + sqrt(10) + sqrt(15) - sqrt(15), "-----", "COLLECTING", "like-terms"), - 4 + sqrt(10) + sqrt(10), highlight(- 4 + 2sqrt(10)))}}}

A = {{{4 + 2sqrt(10)}}}
B = {{{- 4 + 2sqrt(10)}}}

C = AB
C = {{{(4 + 2sqrt(10))(- 4 + 2sqrt(10))}}}
C = {{{- 16 + 8sqrt(10) - 8sqrt(10) + (2sqrt(10))^2}}} ----- FOILing right-side
C = - 16 + 4(10) = - 16 + 40 = 24</font></font></font></b></pre>