Determine the term independent of (3x-1/2x2)6 ?

To determine the term independent of (3x-1/2x^2)^6, we need to find the term that does not contain any x variables.

To do this, we can expand the expression using the binomial theorem. The general form of the binomial theorem is:

(a + b)^n = C(n,0)a^n + C(n,1)a^(n-1)b + C(n,2)a^(n-2)b^2 + ... + C(n,n)b^n

where C(n,k) is the binomial coefficient, which is equal to n! / (k!(n-k)!).

In this case, a = 3x and b = -1/2x^2, and n = 6. We can plug these values into the formula and find the term that does not contain any x variables.

The term independent of x will be the term where the exponent of x is 0. This occurs when k = n in the binomial coefficient formula. So, the term independent of x is:

C(6,6)(3x)^0(-1/2x^2)^6 = C(6,6)(-1/2)^6x^0 = 1 * 1 * (-1/2)^6 = 1 * 1 * 1/64 = 1/64

Therefore, the term independent of (3x-1/2x^2)^6 is 1/64.

The term independent of (3x-1/2x2)6 is the constant term.

Using the binomial theorem, we can expand (3x-1/2x2)6 as:

(3x-1/2x2)6 = 6C0(3x)6 + 6C1(3x)5(-1/2x2) + 6C2(3x)4(-1/2x2)2 + 6C3(3x)3(-1/2x2)3 + 6C4(3x)2(-1/2x2)4 + 6C5(3x)(-1/2x2)5 + 6C6(-1/2x2)6

= 729x6 - 1080x5 + 540x4 - 120x3 + 15x2 - 1.5x + 1/64

Therefore, the term independent of (3x-1/2x2)6 is 1/64.