Simplify the following x 2 + y 3 3 - x 2 - y 3 3

We have to simplify equation x 2 + y 3 3 - x 2 - y 3 3 . We shall use the identity a 3 - b 3 = ( a - b ) a 2 + b 2 + a b . Here, putting the values of a = x 2 + y 3 , b = x 2 - y 3 , in the expression, we get, x 2 + y 3 3 - x 2 - y 3 3 = x 2 + y 3 - x 2 - y 3 x 2 + y 3 2 + x 2 - y 3 2 - x 2 + y 3 x 2 - y 3 = x 2 + y 3 - x 2 - y 3 x 2 + y 3 2 + x 2 - y 3 2 - x 2 + y 3 x 2 - y 3 = x 2 + y 3 - x 2 + y 3 x 2 2 + y 3 2 + 2 x y 6 2 + x 2 2 + y 3 2 - 2 x y 6 2 + x 2 2 - y 3 2 = 2 y 3 x 2 4 + y 2 9 + 2 x y 6 + x 2 4 + y 2 9 - 2 x y 6 + x 2 4 - y 2 9 = 2 y 3 x 2 4 + y 2 9 + 2 x y 6 + x 2 4 + y 2 9 - 2 x y 6 + x 2 4 - y 2 9 By rearranging the variable we get = 2 y 3 x 2 4 + y 2 9 + x 2 4 + x 2 4 = 2 y 3 3 x 2 4 + y 2 9 = x 2 y 2 + 2 y 3 27 . Hence, simplified form of expression x 2 + y 3 3 - x 2 - y 3 3 is x 2 y 2 + 2 y 3 27 .

Simplify the following x + 2 x 3 + x - 2 x 3

We have to simplify the expression x + 2 x 3 + x - 2 x 3 . We shall use the identity a 3 + b 3 = ( a + b ) a 2 + b 2 - a b . Here, putting the values of a = x + 2 x , b = x - 2 x in the expression, we get, x + 2 x 3 + x - 2 x 3 = x + 2 x + x - 2 x x + 2 x 2 + x - 2 x 2 - x + 2 x × x - 2 x = x + 2 x + x - 2 x x × x + 2 x × 2 x + 2 × x × 2 x + x × x + 2 x × 2 x - 2 × x × 2 x - x 2 - 4 x 2 = ( 2 x ) x 2 + 4 x 2 + 4 x x + x 2 + 4 x 2 - 4 x x - x 2 - 4 x 2 = ( 2 x ) x 2 + 4 x 2 + 4 x x + x 2 + 4 x 2 - 4 x x - x 2 + 4 x 2 By rearranging the variable we get, = ( 2 x ) x 2 + 4 x 2 + 4 x 2 + 4 x 2 = 2 x × x 2 + 12 x 2 = 2 x 3 + 24 x . Hence, simplified form of expression x + 2 x 3 + x - 2 x 3 is 2 x 3 + 24 x .

If x 4 + 1 x 4 = 194 , find x 3 + 1 x 3 , x 2 + 1 x 2 and x + 1 x .

We have to find the value of x 3 + 1 x 3 , x 2 + 1 x 2 , x + 1 x . Given x 4 + 1 x 4 = 194 . By adding and subtracting 2 × x 2 × 1 x 2 in left hand side of x 4 + 1 x 4 = 194 we get, x 4 + 1 x 4 + 2 × x 2 × 1 x 2 - 2 × x 2 × 1 x 2 = 194 ⇒ x 4 + 1 x 4 + 2 × x 2 × 1 x 2 - 2 × x 2 × 1 x 2 = 194 ⇒ x 2 + 1 x 2 2 - 2 = 194 ⇒ x 2 + 1 x 2 2 = 194 + 2 ⇒ x 2 + 1 x 2 2 = 196 ⇒ x 2 + 1 x 2 2 = ( 14 ) 2 ⇒ x 2 + 1 x 2 = 14 . Again by adding and subtracting 2 × x × 1 x in left hand side of x 2 + 1 x 2 = 14 , we get, x 2 + 1 x 2 + 2 × x × 1 x - 2 × x × 1 x = 14 ⇒ x + 1 x 2 - 2 × x × 1 x = 14 ⇒ x + 1 x 2 - 2 = 14 ⇒ x + 1 x 2 = 14 + 2 ⇒ x + 1 x 2 = 16 ⇒ x + 1 x 2 = 4 × 4 ⇒ x + 1 x = 4 . Now, cubing on both sides of x + 1 x = 4 , we get, x + 1 x 3 = 4 3 We shall use identity ( a + b ) 3 = a 3 + b 3 + 3 a b ( a + b ) . x 3 + 1 x 3 + 3 × x × 1 x x + 1 x = 4 × 4 × 4 ⇒ x 3 + 1 x 3 + 3 × x × 1 x × 4 = 64 ⇒ x 3 + 1 x 3 + 12 = 64 ⇒ x 3 + 1 x 3 = 64 - 12 ⇒ x 3 + 1 x 3 = 52 . Hence, the values of x 3 + 1 x 3 , x 2 + 1 x 2 , x + 1 x are 52 , 14 , 4 respectively.