Hello from MrBillDoesMath!
Answer:
Your answer is correct. (You have earned the unofficial title of Genius, Jr. );
Discussion:
Whew! This is a long problem. Let me see what I get and compare it to your answer. First each of the 3 terms contains the sqrt(yz) so let's factor that terms out. You equations equals
sqrt(yz) ( sqrt(108) + 3 sqrt(98) + 2 sqrt(75) ) (A)
Hmm.... let's look at the prime factorizations....
108 = 3 * 36 = 3 * (4*9) = 3 * 2^2* 3^2
98 = 2 * 49 = 2 * 7^2
75 = 3 * 25 = 3 * 5^2
So (A) =
sqrt(yz) ( 6 sqrt(3) + 3*7 sqrt(2) + 2*5 sqrt(3) ) =
sqrt(yz) ( ( 6 + 2*5) sqrt(3) + 21 sqrt(2) ) =
sqrt(yz) ( (6 + 10) sqrt(3) + 21 sqrt(2)) =
sqrt(yz) ( 16 sqrt(3) + 21 sqrt(2) )
Our answers agree!
Thank you,
MrB