find term independent of y in expansion of,[x^4÷y^3+y^2÷2x]^10

Answer 1

If you consider that LCM(2,3)=6
a = x^4/y^3
b = y^2/2x
You want terms of the form

a^2b^3 or a^4b^6, ...

Since we have a 10th power expansion, the exponents must add to 10, so we are looking for a^4b^6. That is, the 5th term:

C(10,4) a^4 b^6
= 210 x^16/y^12 y^12/64x^6
= (105/32) x^10