8 If $(6-k i)^{2}=27-36 i$, the value of $k$ is
(1) -36
(3) 3
(2) -3
(4) 6

Answer

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Answer

Thus, the value of $k$ is $\boxed{3}$.

Steps

Step 1 ：We are given that $(6-ki)^2 = 27 - 36i$. Let's expand the left side of the equation.

Step 2 ：Expanding, we get $(6-ki)(6-ki) = 36 - 12ki + k^2i^2$.

Step 3 ：Since $i^2 = -1$, we can simplify the expression to $36 - 12ki - k^2$.

Step 4 ：Now, we can equate the real and imaginary parts of the equation: $36 - k^2 = 27$ and $-12k = -36$.

Step 5 ：Solving the second equation for $k$, we get $k = \frac{-36}{-12} = 3$.

Step 6 ：Thus, the value of $k$ is $\boxed{3}$.