Let a^2=16/44 and b^2=(2+sqrt5)^2, where a is a negative real number and b is a positive real number. If (a+b)^3 can be expressed in the simplified form x*sqrty

Question

Let a^2=16/44 and b^2=(2+sqrt5)^2, where a is a negative real number and b is a positive real number. If (a+b)^3 can be expressed in the simplified form x*sqrty/z where x, y, and z are positive integers, what is the value of the sum x+y+z?

 

Note: sqrt=square root

Answer 1

We have a^2 = 16/44 = 4/11, so a = -2sqrt(11)/11 (since a is negative).

We also have b^2 = (2+sqrt(5))^2 = 4 + 4sqrt(5) + 5 = 9 + 4sqrt(5), so b = sqrt(9+4sqrt(5)).

Now, we can compute (a+b)^3 as follows:

(a+b)^3 = (-2sqrt(11)/11 + sqrt(9+4sqrt(5)))^3

= (-2sqrt(11)/11 + sqrt(9+4sqrt(5)))^2(-2sqrt(11)/11 + sqrt(9+4sqrt(5)))

= (4/11 - 2sqrt(11)/11 + 9+4sqrt(5) - 2sqrt(11)sqrt(9+4sqrt(5))/11)(-2sqrt(11)/11 + sqrt(9+4sqrt(5)))

= (-2sqrt(11)/11 + 9+4sqrt(5) - 2sqrt(11)sqrt(9+4sqrt(5))/11)(sqrt(9+4sqrt(5))) [since the first term in the product is equal to (-2sqrt(11)/11 + sqrt(9+4sqrt(5))) and the second term is equal to sqrt(9+4sqrt(5))]

= (9sqrt(9+4sqrt(5)) + 4sqrt(5)sqrt(9+4sqrt(5)) - 2sqrt(11)sqrt(9+4sqrt(5)) - 2sqrt(11)*4sqrt(5)) / 11

= ((9 + 4sqrt(5))sqrt(9+4sqrt(5)) - 8sqrt(11)sqrt(9+4sqrt(5))) / 11

= (9sqrt(81+36sqrt(5)) + 4sqrt(5)sqrt(81+36sqrt(5)) - 8sqrt(11)sqrt(81+36sqrt(5))) / 11

= ((9+4sqrt(5)-8sqrt(11))sqrt(81+36sqrt(5))) / 11

Thus, we have x = 9+4sqrt(5)-8sqrt(11), y = 81, and z = 11, so x+y+z = 9+4sqrt(5)-8sqrt(11) + 81 + 11 = 101+4sqrt(5)-8sqrt(11). Therefore, the answer is x+y+z = 101+4sqrt(5)-8sqrt(11).