Find the cube root of 10 + 6 3

Let ( 10 + 6 3 ) 1 3 = x + y . . . ( i ) Then ( 10 - 6 3 ) 1 3 = x - y . . . ( ii ) Multiplying i and ii , we get ( 100 - 108 ) 1 3 = x 2 - y ⇒ - 8 1 3 = x 2 - y ⇒ x 2 - y = - 2 ∴ y = x 2 + 2 . . . ( iii ) Now cubing i on both sides, we have 10 + 6 3 = x 3 + y y + 3 x 2 y + 3 x y Equating rational parts on both sides x 3 + 3 x y = 10 ⇒ x 3 + 3 x x 2 + 2 - 10 = 0 ; ∵ y = x 2 + 2 ⇒ 4 x 3 + 6 x - 10 = 0 By remainder theorem, we have x = 1 ∴ y = 1 2 + 2 = 3 ∴ Required cube root is = 1 + 3

Find the cube root of 10 + 6 3

Let ( 10 + 6 3 ) 1 3 = x + y . . . ( i ) Then ( 10 - 6 3 ) 1 3 = x - y . . . ( ii ) Multiplying i and ii , we get ( 100 - 108 ) 1 3 = x 2 - y ⇒ - 8 1 3 = x 2 - y ⇒ x 2 - y = - 2 ∴ y = x 2 + 2 . . . ( iii ) Now cubing i on both sides, we have 10 + 6 3 = x 3 + y y + 3 x 2 y + 3 x y Equating rational parts on both sides x 3 + 3 x y = 10 ⇒ x 3 + 3 x x 2 + 2 - 10 = 0 ; ∵ y = x 2 + 2 ⇒ 4 x 3 + 6 x - 10 = 0 By remainder theorem, we have x = 1 ∴ y = 1 2 + 2 = 3 ∴ Required cube root is = 1 + 3

Find the cube root of 38 + 17 5

Let ( 38 + 17 5 ) 1 3 = x + y . . . ( i ) Then ( 38 - 17 5 ) 1 3 = x - y . . . ( ii ) Multiplying i and ii , we get ( 38 ) 2 - 17 2 × 5 1 3 = x 2 - y ⇒ ( 1444 - 1445 ) 1 3 = x 2 - y ⇒ - 1 = x 2 - y ∴ y = x 2 + 1 Now cubing i both sides, we get 38 + 17 5 = x 3 + 3 x y + 3 x 2 y + y y Equating rational parts, we get x 3 + 3 x y = 38 ⇒ x 3 + 3 x x 2 + 1 - 38 = 0 ; ∵ y = x 2 + 1 ⇒ 4 x 3 + 3 x - 38 = 0 By remainder theorem, we get, x = 2 ⇒ y = 2 2 + 1 = 5 ∴ Required cube root is 2 + 5 .

Find the cube root of 99 - 70 2

Let ( 99 - 70 2 ) 1 3 = x - y . . . ( i ) ( 99 + 70 2 ) 1 3 = x + y . . . ( ii ) Multiplying i and ii , we get ( 99 - 70 2 ) 1 3 ( 99 + 70 2 ) 1 3 = x 2 - y ⇒ ( 9801 - 9800 ) 1 3 = x 2 - y ⇒ 1 = x 2 - y ⇒ y = x 2 - 1 Now cubing i both sides, we get 99 - 70 2 = x 3 + 3 x y - 3 x 2 y - y y Equating rational parts, we get x 3 + 3 x y = 99 ⇒ x 3 + 3 x x 2 - 1 - 99 = 0 ; ∵ y = x 2 - 1 By remainder theorem, we get, x = 3 ∴ y = 3 2 - 1 = 8 ∴ Required cube root is 3 - 8 or 3 - 2 2 .

Find the cube root of 38 14 - 100 2

Given, 38 14 - 100 2 = - 2 2 ( 50 - 19 7 ) Now, 38 14 - 100 2 1 3 = ( - 2 2 ) ( 50 - 19 7 ) 1 3 = - 2 50 - 19 7 1 3 Now let, ( 50 - 19 7 ) 1 3 = x - y . . . ( i ) and ( 50 + 19 7 ) 1 3 = x + y . . . ( ii ) Multiplying i and ii , we get 2500 - 2527 1 3 = x 2 - y ⇒ - 3 = x 2 - y ∴ y = x 2 + 3 Cubing i both sides, we get 50 - 19 7 = x 3 - 3 x 2 y + 3 x y - y y Equating rational parts, we have x 3 + 3 x y = 50 ⇒ x 3 + 3 x x 2 + 3 - 50 = 0 ; ∵ y = x 2 + 3 ⇒ 4 x 3 + 9 x - 50 = 0 i.e., x = 2 (by remainder theorem) y = 7 Hence, the required cube root is - 2 ( 2 - 7 ) = 14 - 2 2 .

Find the cube root of 54 3 + 41 5

Rewriting the given expression as 54 3 + 41 5 = 3 3 18 + 41 3 × 5 3 ⇒ 54 3 + 41 5 1 3 = 3 3 18 + 41 3 5 3 1 3 = 3 18 + 41 3 5 3 1 3 Now, let 18 + 41 3 5 3 1 3 = x + y . . . i Then, 18 - 41 3 5 3 1 3 = x - y . . . ii Multiplying i and ii , we have 18 2 - 41 3 × 5 3 2 1 3 = x 2 - y ⇒ 324 - 1681 × 5 27 1 3 = x 2 - y ⇒ 343 27 1 3 = x 2 - y ⇒ 7 3 = x 2 - y ⇒ y = x 2 - 7 3 Cubing i both sides, we get 18 + 41 3 5 3 = x 3 + 3 x y + 3 x 2 y + y y Equating rational part, we get x 3 + 3 x y = 18 or x 3 + 3 x x 2 - 7 3 = 18 ; ∵ y = x 2 - 7 3 Solving it for x , we get, x = 2 ⇒ y = 4 - 7 3 = 5 3 Hence, the required cube root is 3 2 + 5 3 = 2 3 + 5

Find the cube root of 135 3 - 87 6 .

Rewriting the given expression as 135 3 - 87 6 = 3 3 ( 45 - 29 2 ) Taking cube root of both the sides, we get 135 3 - 87 6 1 3 = 3 45 - 29 2 1 3 Now let, 45 - 29 2 1 3 = x - y . . . i and 45 + 29 2 1 3 = x + y . . . ii Multiplying i and ii , we get ( 2025 - 1682 ) 1 3 = x 2 - y ⇒ 7 = x 2 - y ∴ y = x 2 - 7 Cubing i both sides, we have 45 - 29 2 = x 3 - 3 x 2 y + 3 x y - y y Equating rational parts, we get x 3 + 3 x y = 45 ⇒ x 3 + 3 x x 2 - 7 - 45 = 0 ; ∴ y = x 2 - 7 ⇒ 4 x 3 - 21 x - 45 = 0 i.e., x = 3 (by remainder theorem) ∴ y = 9 - 7 = 2 Hence, the required cube root is 135 3 - 87 6 1 3 = 3 3 - 2 = 3 3 - 6

Find the square root of a + x + 2 a x + x 2

Let a + x + 2 a x + x 2 = m + n , where m , n > 0 . Squaring both sides, we have a + x + 2 a x + x 2 = m + n + 2 m n Equating rational and irrational parts, we have m + n = a + x . . . i 2 m n = 2 a x + x 2 . . . ii ∴ ( m - n ) 2 = ( m + n ) 2 - 4 m n = ( a + x ) 2 - 2 a x + x 2 = a 2 + 2 a x + x 2 - 2 a x - x 2 = a 2 ∴ m - n = a . . . iii Solving i & iii , we get m = a + x 2 & n = x 2 ∴ Required square root is = a + x 2 + x 2 = 2 a + x 2 + x 2

Find the square root of 2 a - 3 a 2 - 2 a b - b 2

Given, 2 a - 3 a 2 - 2 a b - b 2 = 1 2 2 × 2 a - 2 3 a 2 - 2 a b - b 2 = 1 2 [ ( 3 a + b ) + ( a - b ) - 2 ( 3 a + b ) ( a - b ) ] = 3 a + b 2 2 + a - b 2 2 - 2 3 a + b 2 × a - b 2 = 3 a + b 2 - a - b 2 2 ∴ Required square root is 3 a + b 2 - a - b 2

Find the square root of 1 + a 2 + 1 + a 2 + a 4 1 2

Given, 1 + a 2 + 1 + a 2 + a 4 = 1 2 2 + 2 a 2 + 2 1 + a 2 + a 4 = 1 2 1 + a 2 + a + 1 + a 2 - a + 2 1 + a 2 + a 1 + a 2 - a = 1 + a 2 + a 2 + 1 + a 2 - a 2 2 ∴ Required square root is 1 + a 2 + a 2 + 1 + a 2 - a 2