Question

$\sqrt{7 + 4 \sqrt{3} } \: \: \: \: x + 1 \div x$
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Answer 1

Step-by-step explanation:

I think question is like this

If x = √ (7 + 4 √3 ) then find (x + 1/x )

Solution--->

x = √ (7 + 4√3 )

= √{ 4 + 3 + 2 ( 2 ) (√3) }

= √ {(2)² + (√3)² + 2 (2) (√3 ) }

We have an identity

a² + b² + 2 a b = ( a + b )², applying it here

= √ ( 2 + √3 )²

x = (2 + √3 )

Now

1 / x = 1 / (2 + √3 )

multiplying in numerator and denominator by conjugate of denominator which is (2 - √3 ) we get

= (2 - √3 ) / (2 + √ 3 ) ( 2 - √3 )

We have an identity

a² - b² = ( a + b ) ( a - b )

Applying it in denominator we get

= ( 2 - √3 ) / { ( 2 )² - ( √3 )² }

= (2 - √3 ) / (4 - 3 )

= (2 - √3 ) / 1

= ( 2 - √3 )

Now

x + 1/x = ( 2 + √3 ) + ( 2 - √3 )

+(√3 )and (- √3 ) cancel out each other and we get.

= 2 + 2

= 4

Answer 2

Answer:

Step-by-step explanation:

x = √7 + 4√3

1/x = 1/(√7 + 4√3)

1/x = (√7 - 4√3) / (√7 + 4√3) (√7 - 4√3)

1/x = (√7 - 4√3)/ ((√7)^2 - (4√3)^2)

1/x = (√7 - 4√3) / (7 -48)

1/x = -((√7 - 4√3)/41)

x + 1/x = (√7 + 4√3) + (-(√7 - 4√3)/41)

= (√7 + 4√3) - (√7 - 4√3)/41

= (41(√7 + 4√3) - (√7 - 4√3))/41

= {41√7 - √7 + 164√3 + 4√3}/41

= (40√7 + 168√3)/41

                ....................