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2015-04-23T17:11:30+05:30

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Sum of roots, a+b=2
product of roots, ab=4
a^2+b^2=(a+b)^2-2ab=2^2-2(4)=-4

x^2-2x+4=0\implies x^2=2x-4
Since a,b are roots, for k>2 we have :
a^2=2a-4\implies a^k=2a^{k-1}-4a^{k-2}
b^2=2b-4\implies b^k=2b^{k-1}-4b^{k-2}

Add them and get a recurrence relation :
a^k+b^k=2(a^{k-1}+b^{k-1})-4(a^{k-2}+b^{k-2})

Plugin k=3,4,5,\ldots successively in above recurrence relation :a^3+b^3=2(a^2+b^2)-4(a^1+b^1)=2(-4)-4(2)=-16a^4+b^4=2(a^3+b^3)-4(a^2+b^2)=2(-16)-4(-4)=-16a^5+b^5=2(a^4+b^4)-4(a^3+b^3)=2(-16)-4(-16)=32a^6+b^6=2(a^5+b^5)-4(a^4+b^4)=2(32)-4(-16)=128a^7+b^7=2(a^6+b^6)-4(a^5+b^5)=2(128)-4(32)=128a^8+b^8=2(a^7+b^7)-4(a^6+b^6)=2(128)-4(128)=-256a^9+b^9=2(a^8+b^8)-4(a^7+b^7)=2(-256)-4(128)=-1024
1 5 1
This answer is like that one of Einstein's
ikr lol
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2015-04-23T20:41:39+05:30

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Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
              x² - 2 x + 4 = 0
       Given that    a and b are the roots.
   =>  a² - 2 a + 4 = 0        and      b² - 2 b + 4 = 0

Using complex numbers method

   a = [ 2 + √(4-16) ] / 2 = 1 + i √3
      = 2 (Cos π/3 + i Sin π/3 ) = 2 e^{i\frac{\pi}{3}}

   b = [ 2 - √(4-16) ] / 2 = 1 - √3  i
      = 2 (Sin π/3 -  i Sin π/3 ) = 2 e^{-i\frac{\pi}{3}}

a^9 + b^9 = 2^9 * [ e^{i\frac{9\pi}{3} } + e^{i\frac{-9\pi}{3} } ]

So  a⁹+b⁹
       = 2⁹ * [ Cos 3π + i Sin 3π + Cos 3π - i Sin 3π ]
       =  2⁹ * 2 * (-1)
         = - 1024
=====================================
Another simpler method:
              x² - 2 x + 4 = 0

So sum of roots:   a + b = 2      and      product of roots:   a b = 4

  a³ + b³ = (a+b)³ - 3 a b (a+b)        ---- (1)
            =  8 - 3 * 4 * 2 = - 16

  a⁹ + b⁹ = (a³ + b³)³ - 3 a³ b³ (a³ + b³)            applying the same principle as in (1)
             = (-16)³ - 3 (4)³ (-16)
             = - 16³  + 3 * 4 * 16²
             = 16² * [ -16 + 12 ]
             = - 1024

2 5 2
thanks i wanted the second method only :0
**:)