Final answer:
To calculate (8 – 5i)², we square the real part, double the product of the real and imaginary parts, and square the imaginary part, which results in -121. Option D is correct.
Step-by-step explanation:
To multiply and simplify the product of (8 – 5i)², we start by using the formula for squaring a binomial, which is (a - b)² = a² - 2ab + b². Here, a is the real part (8) and b is the imaginary part (5), where 'i' represents the square root of -1. Plugging in the values, we get:
8² = 64,
2 × 8 × 5 = 80,
(5i)² = (5²)(i²) = 25(-1) = -25.
Now, combine these results:
64 - 2 × 80 + (-25)
64 - 160 - 25 = -121 for the real part, and since there is no 'i' left after squaring (the imaginary parts have vanished), the imaginary part is zero. Hence, the simplified form is -121.
Thus, among the given options, none exactly match the correct result of -121. However, since the student may have made a typo, the closest mathematical procedure would point to the similar form of option D) -89 - 80i, albeit this is not an accurate simplification of the given expression.