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Rational Rigor (Posted on 2011-03-22) Difficulty: 3 of 5
Determine the total number of individual rational terms in the multinomial expansion of each of the following expressions:

(I) (√12 + √32 + √48 + √50)6

(II) (√12 + √32 + √48 + √50)8

Hence or otherwise, determine the total number of individual rational terms in the multinomial expansion of (√12 + √32 + √48 + √50)2n in terms of n, whenever n is a positive integer.

No Solution Yet Submitted by K Sengupta    
Rating: 3.5000 (2 votes)

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Possible solution. | Comment 3 of 4 |

I'm not too sure I follow the problem, but here goes:

(12^(1/2) + 32^(1/2) + 48^(1/2) + 50^(1/2))=(2*3^(1/2)+4*2^(1/2)+4*3^(1/2)+5*2^(1/2))

let 2=a, let 3=b

(2*b^(1/2)+4*a^(1/2)+4*b^(1/2)+5*a^(1/2))^2=108*a^(1/2)*b^(1/2)+81a+36b, with 2 rational terms.

(2*b^(1/2)+4*a^(1/2)+4*b^(1/2)+5*a^(1/2))^4=17496 a^(3/2) b^(1/2)+6561a^2+7776a^(1/2)b^(3/2)+17496ab+1296 b^2, with 3 rational terms.

(2*b^(1/2)+4*a^(1/2)+4*b^(1/2)+5*a^(1/2))^6=531441a^3 +2125764a^(5/2) b^(1/2) + 3542940 a^2b + 3149280a^(3/2) b^(3/2) + 1574640ab^2 + 419904 a^(1/2)b^(5/2) + 46656b^3, with 4 rational terms.

(2*b^(1/2)+4*a^(1/2)+4*b^(1/2)+5*a^(1/2))^8=229582512 a^(7/2)b^(1/2)+714256704a^(5/2)b^(3/2)+317447424a^(3/2) b^(5/2)+43046721a^4+535692528a^3b+595213920a^2 b^2+20155392a^(1/2)b^(7/2)+105815808ab^3+1679616 b^4 with 5 rational terms.

So (12^(1/2) + 32^(1/2) + 48^(1/2) + 50^(1/2))^2n will have (n+1) rational terms.

Edited on March 23, 2011, 9:01 am
  Posted by broll on 2011-03-23 08:54:12

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