China | Math Olympiad | Nice Algebra Radical Problem

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[@shmuelzehavi4940]

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.

[@MasterTMathsClass]

Well done 🎉

[@RyanLewis-Johnson-wq6xs]

(Sqrt(6)+Sqrt(2)/Sqrt(8))^9=8.28125+4.78125Sqrt(3)

[@amaama4140]

Very nice math solution, but I genuinely thought this was recorded in 2006. You should consider using better equipment. It really makes a difference.

[@johnmaulkin-ur8lm]

Many thanks; good health to you.

[@MasterTMathsClass]

🙏

[@JutroToDzis]

((sqrt,(3)+1)³)³ (a+b)³=a³+3a²b+3ab²+b³ twice. without x. Simple.

[@shannonmcdonald7584]

Very nice. 2 things I didn't understand: root 2 was replaced with 1, and 4x² was rewritten as (4x + 2).

[@therrontelford2786]

I noticed that too

[@shannonmcdonald7584]

@@therrontelford2786 I figured out the 4x+2, but not the 1

[@evgeniya-nes]

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

[@PauloSustelo]

Newton's binomial is much easier and direct

[@narineaghajanyan702]

Շատ բարդ է լուծած , կարելի արմատ 2-ով կրճատելուց հետո, 2անգամ խորանարդ բարձրացնել։

[@elenaokon2736]

It's interesting, but maybe the direct calculation was even simpler. But anyway, nice exercise!

[@MasterTMathsClass]

Thanks 🎉

[@VanThanhNguyen-zd9gd]

You solve that mathematic become complex

[@robertosierrai7486]

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

[@MasterTMathsClass]

Thanks 🎉

[@nikolayguzman331]

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.

[@MasterTMathsClass]

Nice one🎉

[@TioCelo72]

16*4x^2 is not equal to 16(4x+2).

[@obobro]

4x^2 = 4x+2 from above, that was just a simple replacement 3:30

[@user-wr8dz1mk8o]

Compared to direct calculation, your way takes triple efforts at least.

[@ludmilak9396]

Показатель в выражении -- это буква q или цифра 9? Научитесь, пожалуйста, грамотно писать.

[@shmuelzehavi4940]

No need to ofense !

[@AndriiBilous]

Душнила!😂

[@key_board_x]

= { (√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

[@giovanbattistamelluso3413]

I did not understand anything !

[@JonathanAlabi]

Watch it again

[@shmuelzehavi4940]

Are you joking ?