Simplify A Radical | Sixth Roots

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[@yoav613]

Nice

[@FisicTrapella]

TKALSS 😁

[@phill3986]

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[@williamperez-hernandez3968]

Proving that the correct answer is the golden ratio, phi, can be done quickly. Fact: phi^n = (Ln + Fn sqrt(5))/2, Ln and Fn are Lucas and Fibonacci numbers. Since L6 = 18, F6 = 8, phi^6 = 9 + 4sqrt(5), and we are done.

[@blue_sand6854]

At 2:43: why?

[@walterwen2975]

Simplify A Radical: ⁶√(9 + 4√5) = ?
⁶√(9 + 4√5) = ⁶√[2² + 2(2)(√5) + (√5)²] = ⁶√[(2 + √5)²] = ³√(2 + √5)
8(2 + √5) = 16 + 8√5 = 1 + 3√5 + 15 + 5√5 = 1 + 3√5 + 3(5) + [(√5)²]√5
= 1 + 3√5 + 3(√5)² + (√5)³ = (1 + √5)³, 2 + √5 = (1 + √5)³/8 = [(1 + √5)/2]³
⁶√(9 + 4√5) = ³√(2 + √5) = ³√{[(1 + √5)/2]³} = (1 + √5)/2
Answer check:
⁶√(9 + 4√5) = ⁶√(17.944) = 1.618, (1 + √5)/2 = 1.618; Confirmed
The calculation was achieved on a smartphone with a standard calculator app
Final answer:
Simplified form; ⁶√(9 + 4√5) = (1 + √5)/2