Rationalise the denominator of the following (i) `2/(3sqrt3)`, (ii) `sqrt40/sqrt3`,(iii)`(3+sqrt2)/(4sqrt2)``(iv) 16/(sqrt41-5)`,(v)`(2+sqrt3)/(2-sqrt3)`, (vi)`sqrt6/(sqrt2+sqrt3)` (vii)`(sqrt3+sqrt2)/(sqrt3-sqrt2)`,(viii)`(3sqrt5+sqrt3)/(sqrt5-sqrt3)` , (ix)`(4sqrt3+5sqrt2)/(sqrt48+sqrt18)`

Explanation

(i) Let `E=2/(3sqrt3)`
for raionalising the denominator , multiplying numberator and denominator by `sqrt3`
`E=2/(3sqrt3)xxsqrt3/sqrt3`
`(2sqrt3)/(3xx3)=(2sqrt3)/9`
Let`E=(sqrt40)/(sqrt3)`
For rationalising ther denominator, multiplying numertor and denominator by `sqrt3`
`E=sqrt40/sqrt3xxsqrt2/sqrt2=(3sqrt2+(sqrt2)^(2))/(4(sqrt2)^(2))`
`(3sqrt2+2)/(4xx2)=(3sqrt2+2)/8`
Let `E=(3+sqrt2)/(4sqrt2)`
For rationalising the denominator , multipliting numerator and denominator by `sqrt41+5`
`E=16/(sqrt41-5)xx(sqrt41+5)/(sqrt41+5)`
`(16(sqrt41+5))/((sqrt41)^(2)-(5)^(2))" " [" using identity", (a-b)(a+b)=a^(2)-b^(2)]`
`(16(sqrt41+5))/(41-25)`
`(16(sqrt41+5))/16=sqrt41+5`
(v) Let` E=(2+srt3)/(2-sqrt3)`
For rationalising denominator, multiplying numerator and denominator by `2+sqrt3`
`E=(2+sqrt3)/(2-sqrt3)xx(2+sqrt3)/(2+sqrt3)=((2+sqrt3)^(2))/((2)^(2)-(sqrt3)^(2))`
[ using identity , `(a-b) (a+b)=a^(2)-b^(2)`]
`(2^(2)+(sqrt3)^(2)+2xx2xxsqrt3)/(4-3)`
[ using identity, `(a+b)^(2)=a^(2)+2ab+b^(2)`]
`4+3+4sqrt3=7=4sqrt3`
Let `E=sqrt6/(sqrt2+sqrt3)`
For rationalising the denominator , mutilplying and denominator by `sqrt2-sqrt3`,
`E=(sqrt6)/(sqrt2+sqrt3)xx(sqrt2-sqrt3)/(sqrt2-sqrt3)=(sqrt6(sqrt2-sqrt3))/((sqrt2)^(2)-(sqrt3)^(2))`
[ using identity , `(a-b) (a+b) =a ^(2)-b^(2)]`
`(sqrt6(sqrt2-sqrt3))/(2-3)=sqrt6(sqrt2-sqrt3)/-1=sqrt6(sqrt3-sqrt2)`
`sqrt18-sqrt12=sqrt(9xx2)-sqrt(4xx3)=3sqrt2-2sqrt3.`
Let `E = (sqrt3+sqrt2)/(sqrt5-sqrt3)`
For rationalising the denominator , multiplying numerator and denominator by `sqrt5-sqrt3`,
`E=(sqrt3+sqrt2)/(sqrt3-sqrt2)xx(sqrt3+sqrt2)/(sqrt3+sqrt2)=(sqrt3+sqrt2)^(2)/((sqrt3)^(2)-(sqrt2)^(2))`
[ using identity , `(a+b) (a-b)=a^(2)-b^(2)]`
`((sqrt3)^(2)+(sqrt(2))^(2)+2sqrt3sqrt2)/(3-2)`
[ using identity , `(a +b)^(2)=a^(2)+b^(2)+2ab]`
`3+2+2sqrt6=5+2sqrt6`
(vii) Let `E= (3sqrt5+sqrt3)/(sqrt5-sqrt3)`
For rationalising the denominator , multiplying numerator and denominator by `sqrt5+sqrt3`
`E= (3sqrt5+sqrt3)/(sqrt5-sqrt3)xx(sqrt5+sqrt3)/(sqrt5+sqrt3)=(3sqrt5(sqrt5+sqrt3)+sqrt3(sqrt5+sqrt3))/((sqrt5)^(2)-(sqrt3)^(2))`
[ using identity , `(a +b) (a-b)=a^(2)-b^(2)]`
`(15+3sqrt15+sqrt15+3)/(5-3)=(18+4sqrt15)/2=9+2sqrt15`
(ix) LetE= `(4sqrt3+5sqrt2)/(sqrt48+sqrt18)=(4sqrt3+5sqrt2)/(sqrt(16xx3)+sqrt(9xx2))=(4sqrt3+5sqrt2)/(4sqrt3+3sqrt2)`
For rationalising the denominator, multiping numertor and denominator by `4sqrt3-3sqrt2`
`(4sqrt3+5sqrt2)/(4sqrt3+3sqrt2)xx ((4sqrt3+5sqrt2))/((4sqrt3-sqrt2))`
`(4sqrt3(4sqrt3-3sqrt2)+5sqrt2(4sqrt3-3sqrt2))/((4sqrt3)^(2)-(3sqrt2)^(2))`
[ using identity , `(a+b) (a-b) = a^(2)-b^(2)`]
`(48-12sqrt6+20sqrt6-30)/30`
`(18+8sqrt6)/30=(9+4sqrt6)/15`