Question

Please answer the following question

Answer 1

Answer:

2√2

Step-by-step explanation:

Given,

       x = 3 + 2√2

⇒ x = 3 + 2√2

⇒ x = 2 + 1 + 2√2

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

Using a^2 + b^2 + 2ab = ( a + b )^2

⇒ x = ( √2 + √1 )^2

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

⇒ 1 / x = 1 / ( √2 + 1 )^2

Multiply and divide by ( √2 - 1 )^2 on right hand side

⇒ 1 / x = ( √2 - 1 )^2 / { ( √2 + 1 )( √2 - 1 ) }^2

Using ( a + b)( a - b ) = a^2 - b^2

⇒ 1 / x = ( √2 - 1 )^2 / { ( √2 )^2 - 1 }

⇒ 1 / x = ( √2 - 1 )^2 / 1

⇒ 1 / x = ( √2 - 1 )^2            ...( 2 )

From ( 1 ) and ( 2 ) :

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

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

From this conclusion :

= > x^( 1 / 2 ) + 1 / x^( 1 / 2 ) = √2 + 1 + √2 - 1

= > x^( 1 / 2 ) + x^( - 1 / 2 ) = 2√2

Answer 2

x ^ ½ = ( 3 + 2√2 ) ^ ½

=> √ x = √ ( 3 + 2√2 )

=> √ x = √ { (√2)² + 1² + 2×√2×1 )}

=> √ x = √ { √2 + 1 } ²

=> x ^ 1/2 = √2 + 1

=> x ^ -1/2 = 1 / x^1/2 = 1 / ( √2 + 1 )

=> rationalizing the denominator ,

=> x ^ -1/2 = √ 2 - 1

now ,

=> x^1/2 - x^-1/2 = √ 2 + 1 - √2 + 1 = 2