Question

The third term of the expansion (1/2x - 2y)^4 when written in standard form is:

a) -8x^3y
b) 8x^3y^3
c) -16x^2y^2
d) 16x^2y^2

Answer 1

The third term of the expansion of (1/2x - 2y)^4 according to the binomial theorem is 6x^2y^2. However, this result does not match any of the options provided, suggesting there may be an error in the question or answer choices.

Step-by-step explanation:

To find the third term of the expansion of (1/2x - 2y)^4, we can use the binomial theorem. The general term in the expansion of (a - b)^n is given by T(r+1) = C(n, r) * a^(n-r) * b^r where C(n, r) is the binomial coefficient n choose r and represents the number of ways to choose r elements from a set of n. For the third term (r = 2), the formula is C(4, 2) * (1/2x)^2 * (-2y)^2.

Calculating this, we get C(4, 2) = 6, and thus the term is 6 * (1/4x^2) * (4y^2) which simplifies to 6 * x^2 * y^2. After simplification, we have 6x^2y^2. However, none of the options provided in the question directly matches this result, indicating a possible error in the answer choices given or the question itself.