This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Convert the complex number z to polar form. Given z = sqrt(3) + i. The rectangular form is z = x + yi, where x = sqrt(3) and y = 1. Calculate the modulus r: r = |z| = sqrt(x^2 + y^2) = sqrt((3))^2 + (1)^2 = sqrt(3 + 1) = sqrt(4) = 2 Calculate the argument : Since x > 0 and y > 0, is in the first quadrant. = (y)/(x) = (1)/(sqrt(3)) = ((1)/(sqrt(3))) = ()/(6) So, z in polar form is z = 2((()/(6)) + i (()/(6))). Step 2: Use De Moivre's Theorem to evaluate z^5. De Moivre's Theorem states that for a complex number z = r( + i ), its n-th power is z^n = r^n((n) + i (n)). For n=5: z^5 = 2^5((5 × ()/(6)) + i (5 × ()/(6))) z^5 = 32(((5)/(6)) + i ((5)/(6))) Step 3: Evaluate the trigonometric functions for (5)/(6). The angle (5)/(6) is in the second quadrant. ((5)/(6)) = -( - (5)/(6)) = -(()/(6)) = -sqrt(3)2 ((5)/(6)) = ( - (5)/(6)) = (()/(6)) = (1)/(2) Step 4: Substitute these values back into the expression for z^5. z^5 = 32(-sqrt(3)2 + i (1)/(2)) z^5 = 32 × (-sqrt(3)2) + 32 × ((1)/(2))i z^5 = -16sqrt(3) + 16i Step 5: Compare the result with the given options. The calculated value for z^5 is -16sqrt(3) + 16i. The options provided in the image are: A. 2 B. 3+i C. 3-i D. 4 None of the given options match the calculated result. The final answer is -16sqrt(3) + 16i.