Question

perform the indicated operations.
$(-6 + \sqrt{-9})^2$
$(-6 + \sqrt{-9})^2 = \square$
(simplify your answer. type your answer in the form $a + bi$.)

Explanation:

Step1: Simplify the imaginary number

$\sqrt{-9} = \sqrt{9 \times (-1)} = 3i$
So the expression becomes $(-6 + 3i)^2$

Step2: Apply the square formula

Use $(a+b)^2 = a^2 + 2ab + b^2$, where $a=-6$, $b=3i$

$$\begin{align*} (-6)^2 + 2(-6)(3i) + (3i)^2 \end{align*}$$

Step3: Calculate each term

$(-6)^2 = 36$, $2(-6)(3i) = -36i$, $(3i)^2 = 9i^2 = 9(-1) = -9$

Step4: Combine like terms

$36 - 9 - 36i = 27 - 36i$

Answer:

$27 - 36i$