Question

1. which expression is equivalent to $(-2xki - 3)^2 - 5i^3$, where $i$ is the imaginary unit?

Explanation:

Step1: Expand the square term

First, expand \((-2xki - 3)^2\) using the formula \((a + b)^2=a^2 + 2ab + b^2\), where \(a=-2xki\) and \(b = - 3\).

$$
LATEXBLOCK0
$$

Since \(i^{2}=-1\), we substitute \(i^{2}\) with \(-1\):

$$
LATEXBLOCK1
$$

Step2: Simplify the \(i^{3}\) term

We know that \(i^{3}=i^{2}\times i=-1\times i=-i\). So, \(-5i^{3}=-5\times(-i) = 5i\)

Step3: Combine the two parts

Now, we combine the result from Step 1 and Step 2:

$$
LATEXBLOCK2
$$

(Note: Assuming there is a typo and the original expression is \((-2ki - 3)^2-5i^{3}\) (removing the extra \(x\) as it might be a typo), let's re - calculate with \(x = 1\) for simplicity. If the expression is \((-2ki-3)^2-5i^{3}\))

Step1 (revised): Expand the square term

\((-2ki - 3)^2=(-2ki)^2+2\times(-2ki)\times(-3)+(-3)^2=4k^{2}i^{2}+12ki + 9\)
Substitute \(i^{2}=-1\): \(4k^{2}\times(-1)+12ki + 9=-4k^{2}+12ki + 9\)

Step2 (revised): Simplify the \(i^{3}\) term

\(-5i^{3}=-5\times(-i)=5i\)

Step3 (revised): Combine the two parts

\((-2ki - 3)^2-5i^{3}=-4k^{2}+12ki + 9+5i=-4k^{2}+(12k)i + 5i+9=-4k^{2}+(12k + 5)i+9\)

Answer:

If the original expression is \((-2ki - 3)^2-5i^{3}\) (after correcting the typo), the equivalent expression is \(-4k^{2}+(12k + 5)i + 9\). If the \(x\) is intentional, the equivalent expression is \(-4x^{2}k^{2}+12xki+5i + 9\)