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Verified answer
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