((sqrt,(3)+1)³)³ (a+b)³=a³+3a²b+3ab²+b³ twice. without x. Simple.
х=(√3+1)/2
х^2=(√3+2)/2
х^4=(4√3+7)/4
х^8=(56√3+97)/16
х^9=(153√2+265)/32
Very nice. 2 things I didn't understand: root 2 was replaced with 1, and 4x² was rewritten as (4x + 2).
Very nice math solution, but I genuinely thought this was recorded in 2006. You should consider using better equipment. It really makes a difference.
Aplicando el triángulo de Pascal
You solve that mathematic become complex
It's interesting, but maybe the direct calculation was even simpler. But anyway, nice exercise!
(Sqrt(6)+Sqrt(2)/Sqrt(8))^9=8.28125+4.78125Sqrt(3)
Gracias a estos ejercicios los alumnos se alejan de las matemáticas. Ejercicios que no tienen un objetivo, ni aplicación práctica. Los jóvenes de hoy en día aprenden diferente a los jóvenes que fuimos nosotros. El chico de hoy agarra la calculadora y resuelve en 2 minutos
Շատ բարդ է լուծած , կարելի արմատ 2-ով կրճատելուց հետո, 2անգամ խորանարդ բարձրացնել։
The idea is very deep, but if ((((SQRT 3 +1)/2) ^2)^2)^2 and Result multiply by (SQRT 3 +1)/2) the result will be more faster, for my opinion.
Newton's binomial is much easier and direct
Compared to direct calculation, your way takes triple efforts at least.
La verdad me confunde
= { (√6 + √2) / √8 }^(9)
= { [(√6 + √2).√8] / 8 }^(9)
= { (√48 + √16) / 8 }^(9)
= { (4√3 + 4) / 8 }^(9)
= { (1 + √3) / 2 }^(9)
= { (1 + √3) }^(9) / 2^(9)
First
= (1 + √3)^(9)
= (1 + √3)².(1 + √3)².(1 + √3)².(1 + √3)².(1 + √3)
= (1 + 2√3 + 3).(1 + 2√3 + 3).(1 + 2√3 + 3).(1 + 2√3 + 3).(1 + √3)
= (4 + 2√3).(4 + 2√3).(4 + 2√3).(4 + 2√3).(1 + √3)
= 16.(2 + √3).(2 + √3).(2 + √3).(2 + √3).(1 + √3)
= 16.(4 + 4√3 + 3).(4 + 4√3 + 3).(1 + √3)
= 16.(7 + 4√3).(7 + 4√3).(1 + √3)
= 16.(49 + 56√3 + 48).(1 + √3)
= 16.(97 + 56√3).(1 + √3)
= 16.(97 + 97√3 + 56√3 + 168)
= 16.(265 + 153√3)
= (265 + 153√3) * 2^(4)
Recall
= { (1 + √3) }^(9) / 2^(9)
= (265 + 153√3) * 2^(4) / 2^(9)
= (265 + 153√3) / 2^(5)
= (265 + 153√3) / 32
≈ 16.5626
Показатель в выражении -- это буква q или цифра 9? Научитесь, пожалуйста, грамотно писать.
Nice solution and explanation.
There is another way to solve the problem after finding out that: 2x - 1 = √3 and therefore by squaring:
2x^2 = 2x + 1
By multiplying by x we obtain:
2x^3 = 2x^2 + x = 3x + 1
By taking to the third power we obtain:
8x^9 = 27x^3 + 1 + 9x (3x + 1) = 27x^3 + 27x^2 + 9x + 1
Therefore:
16x^9 = 27(2x^3 ) + 27(2x^2 ) + 9(2x) + 2 = 27(3x + 1) + 27(2x + 1) + 18x + 2
And therefore:
16x^9 = 153x + 56
The rest is known.
Well done 🎉