`root3(xy^2)` ÷ `sqrt(x^2y)` = ??

Assuming that x, y, z are positive real numbers, simplify each of the following . (i) sqrt(x^(3)y^(-2)) (ii) (x^(-2//3) y^(-1//2))^(2) (iii) ((x^(-4))/(y^(-10)))^((5)/(4)) (iv) (root4(81 x^(8)y^(8)z^(16))) div (root3(27x^(3)y^(6)z^(9))) (v) root5(x^(4) root4(x^(3)root3(x^(2)sqrt(x)))) (vi) ((sqrt(x))^(-(2)/(3)) sqrt(y^(4))) div (sqrt(xy^(-(1)/(2))) sqrt(x^(-2)y^(3)))

Square root of x + y + z + 2sqrt(xy + yz is

(sqrt(x)+sqrt(y))^(2)=x+y+2sqrt(xy) and sqrt(x)sqrt(y)=sqrt(xy) , where x and y are positive real numbers . If x=2sqrt(5)+sqrt(3) and y=2sqrt(5)-sqrt(3) , then x^(4)+y^(4) =

(sqrt(x)+sqrt(y))^(2)=x+y+2sqrt(xy) and sqrt(x)sqrt(y)=sqrt(xy) , where x and y are positive real numbers . If a=1+sqrt(2)+sqrt(3) and b=1+sqrt(2)-sqrt(3) , then a^(2)+b^(2)-2a-2b=

The lines represented by the equation x^2 + 2sqrt(3)xy + 3y^(2) -3x -3sqrt(3)y -4=0 , are

lim_(x rarr0)(x sqrt(y^(2)-(y-x)^(2)))/({sqrt(8xy-4x^(2))+sqrt(8xy)}^(3))=