Consider the complex number z=2sqrt2-2sqrt2i. What is z^(4) ? Hint: z has a modulus of 4 and an argument of 315^(@). Choose 1 answer: (A) -11.3-11.3 i (B) -181+181 i (c) -256 (D) -11.3+11.3 i

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Consider the complex number  z=2sqrt2-2sqrt2i. What is  z^(4) ? Hint:  z has a modulus of 4 and an argument of  315^(@). Choose 1 answer: (A)  -11.3-11.3 i (B)  -181+181 i (c) -256 (D)  -11.3+11.3 i

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Given complex number: We are given the complex number z = 2sqrt2 - 2sqrt2i. We need to find z^(4). First, let's confirm the modulus and argument of z. The modulus of z is |z| = sqrt((2sqrt2)^2 + (-2sqrt2)^2) = sqrt(8 + 8) = sqrt(16) = 4. The argument of z is the angle in the standard position whose terminal side contains the point (2sqrt2, -2sqrt2). This is a 45-degree angle in the fourth quadrant, which corresponds to 315 degrees or 7pi/4 radians.Confirming modulus and argument: Now, we can use De Moivre's Theorem to find z^(4). De Moivre's Theorem states that for a complex number in polar form (r(cos(theta) + i*sin(theta))), (r(cos(theta) + i*sin(theta)))^n = r^n(cos(n*theta) + i*sin(n*theta)). Since we have the modulus r = 4 and the argument theta = 315 degrees, we can apply the theorem to find z^(4).Using De Moivre's Theorem: Applying De Moivre's Theorem, we get: z^(4) = (4(cos(315) + i*sin(315)))^4        = 4^4(cos(4*315) + i*sin(4*315))        = 256(cos(1260) + i*sin(1260)) Since 1260 degrees is equivalent to 360 degrees (full circle) times 3 plus an additional 180 degrees, we can simplify cos(1260) and sin(1260) to cos(180) and sin(180).Applying De Moivre's Theorem: We know that cos(180 degrees) = -1 and sin(180 degrees) = 0. Therefore, z^(4) = 256(cos(180) + i*sin(180))                 = 256(-1 + i*0)                 = 256 * -1                 = -256

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