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Simplify \(\dfrac{\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}}{\sqrt{\sqrt{5} + 1}}\)

 May 14, 2020
 #1
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To make the typing simpler, I'm going to substitue x for sqrt(5).

 

Numerator  =  sqrt(x + 2) + sqrt(x - 2)

Denominator  =  sqrt(x + 1)

 

I'm going to square this fraction, reduce it, and then find the square root to get the answer.

 

Numerator:  [ sqrt(x + 2) + sqrt(x - 2) ]2  =  [ (x + 2) + 2·sqrt(x + 2)·sqrt(x - 2) + (x - 2) ]

       =  [ 2x + 2·sqrt(x + 2)·sqrt(x - 2) ]  

       =  [ 2x + 2·sqrt( x2 - 4 ) ]

But, since x = sqrt(5), x2 = 5:

       =  [ 2x + 2·sqrt( 5 - 4 ) ]

       =  [ 2x + 2·1 ]

       =  [ 2x + 2 ]

       =  2( x + 1 )

 

Denominator:  [ sqrt(x + 1) ]2  =  x + 1

 

Combining:  2( x + 1 ) / ( x + 1 )  =  2

 

Since we squared this fraction, we'll need to find the square root of this to get our answer:  sqrt(2)

 May 14, 2020

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