Consider the complex number z=-(7sqrt2)/(2)-(7sqrt2)/(2)i". " Which of the following complex numbers best approximates z^(3) ? Hint: z has a modulus of 7 and an argument of 225^(@). Choose 1 answer: (A) -14.8-14.8 i (B) 242.5-242.5 i (c) -21 (D) 343 i

Question

Consider the complex number  z=-(7sqrt2)/(2)-(7sqrt2)/(2)i". " Which of the following complex numbers best approximates  z^(3) ? Hint:  z has a modulus of 7 and an argument of  225^(@). Choose 1 answer: (A)  -14.8-14.8 i (B)  242.5-242.5 i (c) -21 (D)  343 i

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Given complex number: We are given the complex number z = -(7sqrt2)/(2) - (7sqrt2)/(2)i. To find z^3, we can use the polar form of the complex number, which is given by z = r(cos(θ) + i sin(θ)), where r is the modulus and θ is the argument of z.Modulus and argument: The modulus of z is given as 7. We can verify this by calculating the modulus: |z| = sqrt[(-7sqrt2/2)^2 + (-7sqrt2/2)^2] = sqrt[2*(7^2/4)] = sqrt[49/2] = 7.Polar form: The argument of z is given as 225 degrees. This corresponds to the point in the complex plane where the angle with the positive x-axis (real axis) is 225 degrees, which is in the third quadrant where both the real and imaginary parts are negative.De Moivre's theorem: Now we can write z in polar form: z = 7(cos(225°) + i sin(225°)). To find z^3, we will use De Moivre's theorem, which states that (r(cos(θ) + i sin(θ)))^n = r^n(cos(nθ) + i sin(nθ)).Applying De Moivre's theorem: Applying De Moivre's theorem to find z^3, we get z^3 = 7^3(cos(3*225°) + i sin(3*225°)) = 343(cos(675°) + i sin(675°)).Simplifying angles: Since the sine and cosine functions are periodic with a period of 360°, we can simplify the angles by subtracting multiples of 360°. So, cos(675°) = cos(675° - 2*360°) = cos(-45°) and sin(675°) = sin(675° - 2*360°) = sin(-45°).Calculating z^3: The cosine and sine of -45 degrees are both -sqrt(2)/2. Therefore, z^3 = 343(-sqrt(2)/2 + i(-sqrt(2)/2)).Multiplying by -sqrt(2)/2: Multiplying 343 by -sqrt(2)/2, we get z^3 = 343 * -sqrt(2)/2 + i(343 * -sqrt(2)/2) = -343/sqrt(2) + i(-343/sqrt(2)).Rationalizing the denominator: To simplify further, we can multiply and divide by sqrt(2) to rationalize the denominator: z^3 = -343/sqrt(2) * sqrt(2)/sqrt(2) + i(-343/sqrt(2) * sqrt(2)/sqrt(2)) = -343*sqrt(2)/2 + i(-343*sqrt(2)/2).Final numerical value: Now we calculate the numerical value: -343*sqrt(2)/2 ≈ -242.5. So, z^3 ≈ -242.5 - 242.5i.

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