Simplify \(\dfrac{\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}}{\sqrt{\sqrt{5} + 1}}\)
+23245 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)
