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- เผยแพร่เมื่อ 28 ส.ค. 2024
- 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!
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E=(√3-1)/16
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).
? = (√3-1)/ 16
Ε=(ριζα3 -1)/16
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
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