Consider the complex number z=-3sqrt3-3i. What is z^(3) ? Hint: z has a modulus of 6 and an argument of 210^(@). Choose 1 answer: (A) -216 i (B) -216 (c) -108+187.1 i (D) 108+187.1 i

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Consider the complex number  z=-3sqrt3-3i. What is  z^(3) ? Hint:  z has a modulus of 6 and an argument of  210^(@). Choose 1 answer: (A)  -216 i (B) -216 (c)  -108+187.1 i (D)  108+187.1 i

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Identify modulus and argument: Identify the modulus and argument of the complex number z. Given z = -3sqrt3 - 3i, we are told that the modulus of z is 6 and the argument is 210 degrees. This information will be used to find z^(3) using the polar form of the complex number.Convert to polar form: Convert the complex number z to its polar form. The polar form of a complex number is given by z = r(cos(θ) + i*sin(θ)), where r is the modulus and θ is the argument. For z, we have r = 6 and θ = 210 degrees. Therefore, the polar form of z is z = 6(cos(210°) + i*sin(210°)).Use De Moivre's Theorem: Use De Moivre's Theorem to find z^(3). De Moivre's Theorem states that (r(cos(θ) + i*sin(θ)))^n = r^n(cos(nθ) + i*sin(nθ)). We want to find z^(3), so we will raise the polar form of z to the power of 3.Calculate z^(3): Calculate z^(3) using De Moivre's Theorem. We have z^(3) = (6(cos(210°) + i*sin(210°)))^3 = 6^3(cos(3*210°) + i*sin(3*210°)) = 216(cos(630°) + i*sin(630°)).Simplify trigonometric functions: Simplify the trigonometric functions. Since 630° is equivalent to 270° (as 630° - 360° = 270°), we can simplify the expression to 216(cos(270°) + i*sin(270°)). The cosine of 270° is 0, and the sine of 270° is -1. Therefore, z^(3) = 216(0 - i) = -216i.

Consider the complex number $z=-3 \sqrt{3}-3 i$ . $\newline$ What is $z^{3}$ ? $\newline$ Hint: $z$ has a modulus of $6$ and an argument of $210^{\circ}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-216 i$ $\newline$ (B) $-216$ $\newline$ (C) $-108+187.1 i$ $\newline$ (D) $108+187.1 i$

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