Question

If x=3+2√2 find the value of √x-1/√x)^3

Answer 1

Answer:

8

Step-by-step explanation:

⇒ x = 3 + 2√2

⇒ x = 2 + 1 + 2√2

Writing 2 as ( √2 )^2 and 1 as 1^2 and 2√2 as 2(1)(√2):

⇒ x = ( √2 )^2 + ( 1 )^2 + 2( 1 )( √2 )

⇒ x = ( √2 + 1 )^2                           { a^2 + b^2 + 2ab = ( a + b )^2 }

⇒ √x = √2 + 1

     Thus,

      1 / √x = 1 / ( √2 + 1 )

  Multiply and divide by √2 - 1 :

  ⇒ 1 / √x = ( √2 - 1 ) / ( √2 + 1 )( √2 - 1 )

  ⇒ 1 / √x = ( √2 - 1 ) / ( 2 - 1 )

  ⇒ 1 / √x = √2 - 1

Hence,

   ⇒ √x - 1 / √x

   ⇒ √2 + 1 - ( √2 - 1 )

   ⇒ √2 + 1 - √2 + 1

   ⇒ 2

Thus,

  ( √x - 1 / √x )^3 = ( 2 )^3

                            = 8