Find a factor which will rationalise 3 3 - 2

Given, 3 3 - 2 Put 3 1 3 = x , 2 1 2 = y , then x 6 = 3 2 & y 6 = 2 3 are both rational and x 6 - y 6 = x 3 - y 3 x 3 + y 3 = ( x - y ) x 2 + y 2 + x y x 3 + y 3 = ( x - y ) x 5 + x 4 y + x 3 y 2 + x 2 y 3 + x y 4 + y 5 The rationalizing factor is x 5 + x 4 y + x 3 y 2 + x 2 y 3 + x y 4 + y 5 = 3 5 3 + 3 4 3 · 2 1 2 + 3 · 2 2 2 + 3 2 3 · 2 3 2 + 3 1 3 2 2 + 2 5 2

Find the factor which will rationalise 5 6 + 2 3 .

Given, 5 6 + 2 3 Put x = 5 1 6 , y = 2 1 3 ∴ x 6 , y 6 both are rational and x 6 - y 6 = x 3 + y 3 x 3 - y 3 = ( x + y ) x 2 - x y + y 2 x 3 - y 3 = ( x + y ) x 5 - x 4 y + x 3 y 2 - x 2 y 3 + x y 4 - y 5 The required factor is, therefore x 5 - x 4 y + x 3 y 2 - x 2 y 3 + x y 4 - y 5 = 5 5 6 - 5 2 3 · 2 1 3 + 5 1 2 · 2 2 3 - 5 1 3 · 2 + 5 1 6 · 2 4 3 - 2 5 3

Find the factor which will rationalise a 1 6 + b 1 4

Given, a 1 6 + b 1 4 Put x = a 1 6 , y = b 1 4 ∴ a = x 6 , b = y 4 x 12 , y 12 are both rational and x 12 - y 12 = x 6 - y 6 x 6 + y 6 = x 3 - y 3 x 3 + y 3 x 6 + y 6 = x + y x 2 - x y + y 2 x 9 + x 3 y 6 + x 6 y 3 + y 9 = x + y x 11 - x 10 y + x 9 y 2 - x 8 y 3 + x 7 y 4 - x 6 y 5 + x 5 y 6 … … … + y 11 Rationalising factor is = x 11 - x 10 y + x 9 y 2 - x 8 y 3 + x 7 y 4 - x 6 y 5 + x 5 y 6 … … … + y 11 = a 11 6 - a 10 6 · b 1 4 + a 9 6 · b 2 4 - a 8 6 · b 3 4 + … + b 11 4

Find the factor which will rationalise 3 3 - 1 .

Given, 3 3 - 1 Put x = 3 1 3 ⇒ x 3 = 3 and y = 1 ⇒ y 3 = 1 , then x 3 , y 3 both are rational. Now, x 3 - y 3 = ( x - y ) x 2 + x y + y 2 ∴ Required rationalising factor is x 2 + x y + y 2 = 3 2 3 + 3 1 3 + 1

Find the factor which will rationalise 2 + 7 4 .

Given, 2 + 7 4 Put x = 2 ⇒ x 4 = 2 4 and y = 7 1 4 ⇒ y 4 = 7 , then x 4 , y 4 both are rational. ∵ x 4 - y 4 = x 2 - y 2 x 2 + y 2 = x + y x - y x 2 + y 2 = ( x + y ) x 3 - x 2 y + x y 2 - y 3 Therefore, the required rationalising factor is x 3 - x 2 y + x y 2 - y 3 ⇒ 2 3 - 2 2 · 7 1 4 + 2 · 7 1 2 - 7 3 4