Nice Square Root Math Simplification Problem

[@NurulAlom-z5t]

Excellent

[@davidseed2939]

this is all about noticing a pattern.
let a=√11+√7 b=√3+√7
so if x= T( top) / B(bottom)
T=ab B=a+b
now define the conjugates of a and b
a’ =√11−√7 aa’=11-7=4
b’=√3−√7 bb’= 3-7=-4
now multiply T and B by a’b’
T=aa’.bb’=-16
B =aa’b’ + bb’a = 4b’ -4a’
= 4( √3−√7) -4(√11−√7 )
=-4(√11−√3)
so x=T/B= -16/(-4(√11−√3)
x=4/(√11−√3)
now rationalise the denominator again using the conjugate.
using (√11−√3 ).(√11+3) =11−3
x=4(√11+√3)/(11-3)
x=(√11+√3)/2
Answer

[@davidseed2939]

11:05 This shows all the working (in reverse )
I think the a and b substitution makes the process clearer

[@nellyjohnson7316]

You made a mistake.