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Simplify \(\sqrt{14 + 6 \sqrt{5}}\)

 

Any idea how to do this?

 Jul 11, 2020
 #1
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Simplify the following:
sqrt(6 sqrt(5) + 14)

Factor 2 out of 14 + 6 sqrt(5) giving 2 (3 sqrt(5) + 7):
sqrt(2 (3 sqrt(5) + 7))

2 (7 + 3 sqrt(5)) = 9 + 6 sqrt(5) + 5 = 9 + 6 sqrt(5) + (sqrt(5))^2 = (sqrt(5) + 3)^2:
sqrt((sqrt(5) + 3)^2)

Cancel exponents. sqrt((3 + sqrt(5))^2) = sqrt(5) + 3:

 3 + sqrt(5)

 Jul 11, 2020

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