Consider the complex number $\newline$ $z=-\frac{5 \sqrt{2}}{2}+\frac{5 \sqrt{2}}{2} i \text { }$ $\newline$ Which of the following complex numbers best approximates $z^{3}$ ? Hint: $z$ has a modulus of $5$ and an argument of $135^{\circ}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-10.6+10.6 i$ $\newline$ (B) $-125$ $\newline$ (C) $-125 i$ $\newline$ (D) $88.4+88.4 i$

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Q. Consider the complex number

\newline

z=-\frac{5 \sqrt{2}}{2}+\frac{5 \sqrt{2}}{2} i \text { }

\newline

Which of the following complex numbers best approximates

z^{3}

? Hint:

z

has a modulus of

5

and an argument of

135^{\circ}

\newline

Choose

1

answer:

\newline

(A)

-10.6+10.6 i

\newline

(B)

-125

\newline

(C)

-125 i

\newline

(D)

88.4+88.4 i

Identify modulus and argument: Identify the modulus and argument of the complex number $z$ . The complex number $z$ is given in the form $z = a + bi$ , where $a = -\frac{5\sqrt{2}}{2}$ and $b = \frac{5\sqrt{2}}{2}$ . The modulus of $z$ is the distance from the origin to the point $(a, b)$ in the complex plane, which can be calculated using the formula $|z| = \sqrt{a^2 + b^2}$ .
Calculate modulus of z: Calculate the modulus of z. $\newline$ $|z| = \sqrt{\left(-\frac{5\sqrt{2}}{2}\right)^2 + \left(\frac{5\sqrt{2}}{2}\right)^2} = \sqrt{\left(\frac{25}{2}\right) + \left(\frac{25}{2}\right)} = \sqrt{25} = 5.$ $\newline$ The modulus of z is indeed $5$ , as given in the hint.
Determine argument of $z$ : Determine the argument of $z$ . $\newline$ The argument of $z$ is the angle $\theta$ made with the positive $x$ -axis. Given that $z$ has an argument of $135$ degrees, we can use this information directly without calculation.
Express $z$ in polar form: Express $z$ in polar form. $\newline$ Using the modulus and argument, we can express $z$ in polar form as $z = |z|(\cos(\theta) + i\sin(\theta))$ . For $z$ , this is $z = 5(\cos(135°) + i\sin(135°))$ .
Calculate $z^3$ using De Moivre's Theorem: Calculate $z^{(3)}$ using De Moivre's Theorem. $\newline$ De Moivre's Theorem states that $(r(\cos(\theta) + i\sin(\theta)))^n = r^n(\cos(n\theta) + i\sin(n\theta))$ . For $z^{(3)}$ , this becomes $z^{(3)} = 5^3(\cos(3\times135°) + i\sin(3\times135°))$ .
Compute value of $z^3$ : Compute the value of $z^{3}$ . $z^{3} = 125(\cos(405°) + i\sin(405°))$ . Since $405°$ is equivalent to $45°$ (as we can subtract full rotations of $360°$ ), we have $z^{3} = 125(\cos(45°) + i\sin(45°))$ .
Simplify expression for $z^3$ : Simplify the expression for $z^{3}$ . $\cos(45°) = \sin(45°) = \sqrt{2}/2$ . Therefore, $z^{3} = 125(\sqrt{2}/2 + i\sqrt{2}/2)$ .
Calculate real and imaginary parts of $z^3$ : Calculate the real and imaginary parts of $z^{3}$ . The real part is $125(\sqrt{2}/2)$ and the imaginary part is $125(\sqrt{2}/2)i$ . Multiplying these out gives us $z^{3} = 125\sqrt{2}/2 + 125\sqrt{2}/2i \approx 88.4 + 88.4i$ .