Math Olympiad l Simplify this Radical if You’re a Genius l 90% Failed to solve!

[@RealQinnMalloryu4]

x^1 2^3 ➖ 3^410/10 2^1 ➖3^2^2 1 ➖ 3^1^2 3^2 (x ➖ 3x+2).

[@walterwen2975]

Math Olympiad: ¹²√(161 - 720/√20) =?
720/√20 = [36(√20)²]/√20 = 4(9)(√20) = 2(9)(2√20) = 2(√81)(√80)
161 - 720/√20 = (√81)² - 2(√81)(√80) + (√80)² = (√81 - √80)² = (9 - 4√5)² > 0
9 - 4√5 = (√5)² - 2(√5)(2) + 2² = (√5 - 2)², (9 - 4√5)² = [(√5 - 2)²]² = (√5 - 2)⁴
8(√5 - 2) = 8√5 - 16 = 5√5 - 15 + 3√5 - 1 = (√5)³ - 3(√5)²(1) + 3(√5)(1)² - 1³ = (√5 - 1)³
(√5 - 2) = [(√5 - 1)³]/8 = [(√5 - 1)/2]³ > 0, (√5 - 2)⁴ = {[(√5 - 1)/2]³}⁴ = [(√5 - 1)/2]¹²
¹²√(161 - 720/√20) = ¹²√[(√5 - 2)⁴] = ¹²√{[(√5 - 1)/2]¹²}= (√5 - 1)/2 > 0

[@adribber]

¹²√(161 - 720/√20) = ?
Let, k = ¹²√(161 - 720/√20)
k = ¹²√(161 - 720√5/2√5*√5)
= ¹²√(161 - 720√5/10)
= ¹²√{161 - 2√(5*6⁴)}
= ¹²√{161 - 2√(80*81)}
= ¹²√{81 - 2√(80*81) + 80}
= ¹²√(√81 - √80)²
= ⁶√{9 - 2√(4*5)}
= ⁶√{5 - 2√(4*5) + 4}
= ⁶√(√5 - √4)²
= ³√(√5 - 2)
Let, a³ = √5 - 2 , b³ = √5 + 2
a³ - b³ = - 4
a³b³ = 1
ab = 1
(a - b)³ = a³ - b³ - 3ab(a - b)
= - 4 - 3(a - b)
(a - b)³ + 3(a - b) + 4 = 0
(a - b)³ + 1 + 3(a - b + 1) = 0
(a - b + 1){(a - b)² - (a - b) + 1} + 3(a - b + 1) = 0
(a - b + 1){(a - b)² - (a - b) + 4} = 0
a - b + 1 = 0 , (a - b)² - (a - b) + 4 = 0
(a - b)² - (a - b) + 4 = 0
D = (- 1)² - 4*1*4 = - 15 < 0 Rejected
a - b + 1 = 0
∴ b = a + 1
Recall, ab = 1
a² + a - 1 = 0
a = [- 1 ± √{1² - 4*1*(- 1)}]/2*1
= (- 1 ± √5)/2
∴ a = (- 1 + √5)/2 (a > 0)
k = ³√(√5 - 2)
= ³√a³
= a
∴ k = (- 1 + √5)/2