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(sqrt(3)-sqrt(2))^10

 Nov 14, 2015
 #1
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   (sqrt(3)-sqrt(2))^10=(5-2sqrt(6))^5

 Nov 14, 2015
 #2
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+5

(sqrt(3)-sqrt(2))^10

 

 

 

 

\((\sqrt3-\sqrt2)^{10}\\ =(-\sqrt2)^{10}+10C1(-\sqrt2)^9(\sqrt3)+10C2(-\sqrt2)^8(\sqrt3)^2+10C3(-\sqrt2)^7(\sqrt3)^3+10C4(-\sqrt2)^6(\sqrt3)^4+10C5(-\sqrt2)^5(\sqrt3)^5+10C6(-\sqrt2)^4(\sqrt3)^6+10C7(-\sqrt2)^3(\sqrt3)^7+10C8(-\sqrt2)^2(\sqrt3)^8+10C9(-\sqrt2)^1(\sqrt3)^9 +(\sqrt3)^{10} \\ =32+10(-16\sqrt2)(\sqrt3)+10C2*16*(\sqrt3)^2+10C3(-8\sqrt2)(\sqrt3)^3+10C4*8(\sqrt3)^4+10C5(-4\sqrt2)(\sqrt3)^5+10C6*4(\sqrt3)^6+10C7(-2\sqrt2)(\sqrt3)^7+10C8*2(\sqrt3)^8+10C9(-\sqrt2)(\sqrt3)^9 +(\sqrt3)^{10} \\ =32+10(-16\sqrt2)(\sqrt3)+10C2*16*3+10C3(-8\sqrt2)(3\sqrt3)+10C4*8*9+10C5(-4\sqrt2)(9\sqrt3)+10C6*4*27+10C7(-2\sqrt2)(27\sqrt3)+10C8*2*81+10C9(-\sqrt2*81\sqrt3) +243 \\ =32+10(-16\sqrt2)(\sqrt3)+10C2*16*3+10C3(-8\sqrt2)(3\sqrt3)+10C4*8*9+10C5(-4\sqrt2)(9\sqrt3)+10C6*4*27+10C7(-2\sqrt2)(27\sqrt3)+10C8*2*81+10C9(-\sqrt2*81\sqrt3) +243 \\ =32+10(-16\sqrt6)+10C2*48+10C3(-24\sqrt6)+10C4*72+10C5(-36\sqrt6)+10C6*108+10C7(-54\sqrt6)+10C8*162+10C9(-81\sqrt6) +243 \\ =32+48*10C2 +72*10C4 +108*10C6 +162*10C8 +243 - [160 +10C3*24+10C5*36+10C7*54+10C9*81]\sqrt6 \\ \)

 

You can simplify it the rest of the way.

If there are no careless errors (which is relatively unlikely) then it should simplify to

 

\(47523-19402\sqrt6\qquad \qquad \mbox{Ref: Wolfram Alpha}\)

 Nov 14, 2015

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