Sol.   `x = ((sqrt(5) + sqrt(3))/(sqrt(5) - sqrt(3))) xx ((sqrt(5) + sqrt(3))/(sqrt(5)+ sqrt(3)))`

               = `((sqrt(5) - sqrt(3))^2)/((5 - 3)) =  (5 + 3 + 2sqrt(15))/(2) = 4 + sqrt(15)`

`y = ((sqrt(5) - sqrt(3))/(sqrt(5) + sqrt(3))) xx ((sqrt(5)- sqrt(3))/(sqrt(5)- sqrt(3)))`

                = `((sqrt(5) - sqrt(3))^2)/ ((5 - 3))   =  (5 + 3 - 2sqrt(15))/(2)     = 4 - sqrt(15)`

`:.`       `x^2 + y^2 = (4 + sqrt(15))^2 + (4 - sqrt(15))^2`

               = `2[(4)^2 + sqrt(15)^2]` =  2 x 31 = 62.

Sol. `(sqrt(2))/((2 + sqrt(2)))`

       = `(sqrt(2))/((2 + sqrt(2)))` x `((2 - sqrt(2)))/((2 - sqrt(2)))`

       = `(2sqrt(2) - 2)/(4 - 2) = (2(sqrt(2) - 1))/(2) `

        = `(sqrt(2) - 1) ` = ( 1.4142 - 1 ) =  0.4142.

Sol.     L.C.M. of 10, 12, 15, 18 =  180

Now , 180 = 2 x 2 x 3 x 3 x 5

                  = `2^2 xx 3^2 xx 5`

To make it a perfect square ,it  must be multiplied by 5.

`:.`      Required number = `(2^2 xx 3^2 xx 5^2)` = 900.

Sol.     `sqrt ( 5/3) = sqrt((5 xx 3)/ (3 xx 3))`

            = `sqrt(15)/3     = 3.88/3 = 1.2933.....  = 1.29overline(3)`

Sol  `x^2 + y^2 = (1 + sqrt(2))^2 + (1 - sqrt(2))^2`

                        =  ` 2[(1)^2 + (sqrt(2))^2]`

                       = 2 x 3 = 6.  

Sol.  Given exp. = `sqrt((12.1 + 8.1) (12.1 - 8.1))/((0.25) (0.25 + 19.95))`

                           = `sqrt((20.2 xx 4)/(0.25 xx 20.2))`

                            = `sqrt(4/0.25) = sqrt(400/25) = sqrt(16) `  = 4

Sol. Given exp. `sqrt(9.5 xx .0085 xx 18.900)/(.0017 xx 1.9 xx 0.021)`

      Now , since the sum of decimal places in the numerator and denominator under the radical sign is the same, we remove the decimal.

`:.`      Given exp.  =` sqrt(95 xx 85 xx 18900)/(17 xx 19 xx 21)`

                                = `sqrt(5 xx 5 xx 900)` = 5 x 30 = 150.

Sol .    `sqrt(192) - 1/2 sqrt(48) - sqrt(75)`

            = `sqrt(64 xx 3) - 1/2 sqrt(16 xx 3) - sqrt(25 xx 3)`

            = ` 8sqrt(3) - 1/2 xx 4 sqrt(3) - 5sqrt(3)`

           = ` 3sqrt(3) - 2sqrt(3)`

           = ` sqrt(3)   `    =  1.732

Sol.    `sqrt(1 + x/144) = 13/12`

`rArr     (1 + x/144) = (13/12)^2`

`rArr      (1 + x/144) = 169/144`

`rArr       x/144 = 169/144 - 1`

`rArr        x/144 = 25/144`

`rArr       x =  25.`

Sol.       `sqrt(0.28900/0.00121`

            `sqrt(28900/121)` = `170/11`