An Intriguing Radical Problem | Algebra Challenge

An Intriguing Radical Problem | Algebra Challenge
Dive into this intriguing radical problem and challenge your algebra skills! In this video, we'll explore a fascinating radical equation that will test problem-solving abilities. Whether you're preparing for a math competition or just love solving algebra problems, this challenge is perfect for you. Watch the video, try to solve the problem, and let us know your solution in the comments below. Happy solving!
🔍 In this video:
Detailed walkthrough of a challenging algebra problem from the Brazilian Math Olympiad.
Tips and tricks for solving complex algebraic equations.
Encouragement to enhance your problem-solving skills and mathematical thinking.
📣 Call to Action:
Have a go at the problem yourself before watching the solution!
Share your solutions and approaches in the comments below.
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Thank You for Watching!!

[@abcekkdo3749]

E=(√3-1)/16

[@user-kp2rd5qv8g]

Let t = 3^1/8. Then, x = (t+1)(t^2+1)(t^4+1) = [(t^4-1)(t^4+1)]/(t-1) = [t^8-1)/(t-1) = 2/(t-1). So, 1/x = (t-1)/2. Thus, f(x)= 1/x^4+2/x^3+3/2x^2 +1/2x = 1/16[t^4-1] = 1/16(√3 -1).

[@tejpalsingh366]

? = (√3-1)/ 16

[@user-ny6jf9is3t]

Ε=(ριζα3 -1)/16

[@user-ee7nw2rx9s]

Искомое выражение
Возьмем 1 и 3 слагаемое, а также 2 и 4
Тогда
(1/х^2)*(1/х^2+3)+(1/х)*(2/х^2+1/2)
Надо только полученное 1/х возвести в квадрат и никаких 4 степеней нет

[@ABHISHEKKUMAR-01024]

We have,
x = (√3 + 1)(⁴√3 + 1)(⁸√3 + 1)
= (²√3 + 1)(⁴√3 + 1)(⁸√3 + 1)
= (3¹/² + 1)(3¹/⁴ + 1)(3¹/⁸ + 1)
= {(3¹/⁸)⁴ + 1} {(3¹/⁸)² + 1} (3¹/⁸ + 1)
= (a⁴ + 1) (a² + 1) (a + 1), where a = 3¹/⁸
= {(a⁴ + 1) (a² + 1) (a + 1) (a - 1) } / (a - 1)
= { (a⁴ + 1) (a² + 1) (a² - 1) } / (a - 1)
= [ (a⁴ + 1) { (a²)² - 1²} ] / (a - 1)
= { (a⁴ + 1) (a⁴ - 1) } / (a - 1)
= { (a⁴)² - 1²} / (a - 1)
= (a⁸ - 1) / (a - 1)
= { (3¹/⁸)⁸ - 1} / (a - 1)
= (3 - 1) / (a - 1)
= 2 / (a - 1)
Therefore,
x = 2 / (a - 1)
or, x (a - 1) = 2
or, a - 1 = 2 / x
or, (2 / x) + 1 = a
or,
{(2 / x) + 1}⁴ = a⁴
or,
(2 / x)⁴ + 4 (2 / x)³ + 6 (2 / x)² + 4 (2 / x)
+ 1 = (3¹/⁸)⁴
or,
(16 / x⁴) + 4 (8 / x³) + 6 (4 / x²) + 4 (2 / x)
+ 1 = 3¹/²
or, (16 / x⁴) + (32 / x³) + (24 / x²)
+ (8 / x) + 1 = √3
or,
(16 / x⁴) + (32 / x³) + (24 / x²)
+ (8 / x) = √3 - 1
or, 8 { (2 / x⁴) + (4 / x³) + (3 / x²)
+ (1 / x) } = √3 - 1
or, (2 / x⁴) + (4 / x³) + (3 / x²)
+ (1 / x) = (√3 - 1) / 8
or, 2 { (1 / x⁴) + (2 / x³) + (3 / (2x²) )
+ (1 / (2x) ) } = (√3 - 1) / 8
or, (1 / x⁴) + (2 / x³) + (3 / (2x²) )
+ (1 / (2x) ) = (1/2) [ (√3 - 1) / 8 ]
= (√3 - 1) / 16