`1/(4 sqrt(3)-3 sqrt(5))`

The value of {1/((sqrt(6) - sqrt(5))) + 1/((sqrt(5) + sqrt(4))) + 1/((sqrt(4) + sqrt(3))) - 1/((sqrt(3) - sqrt(2))) + 1/((sqrt(2) - 1))} is :

Simplify (i) (4+ sqrt(5))/(4-sqrt(5))+(4-sqrt(5))/(4+sqrt(5)) (ii) (1)/(sqrt(3) + sqrt(2)) - (2)/(sqrt(5)-sqrt(3)) -(2)/(sqrt(2) - sqrt(5)) (iii) (2+sqrt(3))/(2-sqrt(3)) + (2-sqrt(3))/(2+sqrt(3)) + (sqrt(3)-1)/(sqrt(3)+1) (iv) (2+sqrt(6))/(sqrt(2)+sqrt(3))+(6sqrt(2))/(sqrt(6)+sqrt(3)) -(8sqrt(3))/(sqrt(6)+sqrt(2))

Arrange the following in decending order sqrt(3) - sqrt(2), sqrt(4) - sqrt(3), sqrt(5) - sqrt(4), sqrt(2) - 1 .

Prove that (i) (1)/(3+sqrt(7)) + (1)/(sqrt(7)+sqrt(5))+(1)/(sqrt(5)+sqrt(3)) +(1)/(sqrt(3)+1)=1 (ii) (1)/(1+sqrt(2))+(1)/(sqrt(2)+sqrt(3))+(1)/(sqrt(3)+sqrt(4))+(1)/(sqrt(4)+sqrt(5))+(1)/(sqrt(5)+sqrt(6))+(1)/(sqrt(6)+sqrt(7)) +(1)/(sqrt(7)+sqrt(8))+(1)/(sqrt(8) + sqrt(9)) = 2

Simplify: 2/(sqrt(5)+\ sqrt(3))+1/(sqrt(3)+\ sqrt(2))-3/(sqrt(5)+\ sqrt(2))

1/(sqrt(2)+sqrt(3)+sqrt(5))+1/(sqrt(2)+sqrt(3)-sqrt(5))

(1)/(sqrt(2)+sqrt(3)-sqrt(5))+(1)/(sqrt(2)-sqrt(3)-sqrt(5))

Prove that sin 9^circ = 1/4 (sqrt(3 + sqrt5) - sqrt(5 - sqrt5)) .

If x = (sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3)) , find the value of x^(3) + (1)/(x^(3)) .