41. Expand (0.4a + 0.5b)^2

Solution:

We need to expand the expression (0.4a + 0.5b)^2. This is a perfect square trinomial of the form (x + y)^2 = x^2 + 2xy + y^2.

In this case, x = 0.4a and y = 0.5b.

1. Square the first term: (0.4a)^2 = (0.4)^2 * a^2 = 0.16a^2
2. Multiply the two terms and then by 2: 2 * (0.4a) * (0.5b) = 2 * (0.20ab) = 0.4ab
3. Square the second term: (0.5b)^2 = (0.5)^2 * b^2 = 0.25b^2

Combining these, we get:

(0.4a + 0.5b)^2 = 0.16a^2 + 0.4ab + 0.25b^2

Answer: 0.16a^2 + 0.4ab + 0.25b^2