Question

find the greatest term in the expansion of (1+x) raised to thepower of10, x=2/3

Answer 1

Hmmm. seems like a pointless waste of time. Any exercise to see whether you understand Pascal's Triangle could involve much less work.

Anyway, the coefficients of (1+x)^10 are

1 10 45 120 210 252 210 120 45 10 1
The trailing terms get less and less when x<1, so we just have to look at the first 6 terms:

1 10*2/3 45*4/9 120*8/27 210*16/81 252*32/243

1 6.66 20 35.55 41.48 33.18

we have a winner: 210*16/81