Question

simplify.

\\(-8 + 5i)^2\\

Explanation:

Identify the expression and expansion formula

We need to simplify the square of a complex binomial:

$$ (-8 + 5i)^2 $$

We will use the binomial expansion formula:

$$ (a + b)^2 = a^2 + 2ab + b^2 $$

where \(a = -8\) and \(b = 5i\).

Apply the binomial expansion formula

Substitute \(a = -8\) and \(b = 5i\) into the formula:

$$ (-8 + 5i)^2 = (-8)^2 + 2(-8)(5i) + (5i)^2 $$

Simplify each term individually

Calculate the value of each term in the expansion:

$$ (-8)^2 = 64 $$

$$ 2(-8)(5i) = -80i $$

$$ (5i)^2 = 25i^2 $$

Substitute the value of the imaginary unit

Recall that the imaginary unit \(i\) satisfies:

$$ i^2 = -1 $$

Substitute this into the third term:

$$ 25i^2 = 25(-1) = -25 $$

Combine the real and imaginary parts

Combine the real numbers and keep the imaginary term:

$$ 64 - 80i - 25 = (64 - 25) - 80i = 39 - 80i $$

Answer:

\(39 - 80i\)