Let `sqrt(7+4sqrt(3))=sqrt(x)+sqrt(y)`
Squaring both the sides,
`7+4sqrt(3)=x+y+2sqrt(xy)`
`implies x+y=7 and sqrt(xy)=2sqrt(3) implies xy 12`
By solving , we get x=4 and y=3
`sqrt(x)+sqrt(y)=sqrt(4)+sqrt(3)=2+sqrt(3)`
(b) Let the given expression be equal to `a+sqrt(x)+sqrt(y)+sqrt(z)`
As per the method discussed , `a=10, b=24, c=60 and d=40 `
`x=(1)/(2)sqrt((bd)/(c))=(1)/(2)sqrt((24xx40)/(60))=2`
`y=(1)/(2)sqrt((bc)/(d))=(1)/(2)sqrt((24xx60)/(40))=3`
`z=(1)/(2)sqrt((cd)/(b))=(1)/(2)sqrt((60xx40)/(24))=5`
`therefore sqrt(x)+sqrt(y)+sqrt(z)=sqrt(2)+sqrt(3)+sqrt(5)`
Alternative method:
`=sqrt(10+sqrt(24)+sqrt(60)+sqrt(40))`
`=sqrt(10+2sqrt(6)+2sqrt(15)+2sqrt(10))`
`=sqrt((2+3+5)+2sqrt(2(3))+2sqrt(3(5))=2sqrt(2(5)))`
`=sqrt((2)+sqrt(3)+sqrt(5))^(2)`
`=sqrt(2)+sqrt(3)+sqrt(5)`
