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: Identify the complex numbers a, b, and c in polar form. The given complex numbers are: a = sqrt(2) ((()/(6)) + i (()/(6))) b = sqrt(2) ((()/(4)) + i (()/(4))) c = sqrt(2) ((()/(3)) + i (()/(3))) These are in the form r( + i ), where r = sqrt(2) for all three. Step 2: Apply De Moivre's Theorem to calculate a^6. De Moivre's Theorem states that (r( + i ))^n = r^n((n) + i (n)). For a^6: a^6 = (sqrt(2))^6 ((6 · ()/(6)) + i (6 · ()/(6))) a^6 = 2^3 (() + i ()) a^6 = 8 (-1 + i · 0) a^6 = -8 Step 3: Apply De Moivre's Theorem to calculate b^6. For b^6: b^6 = (sqrt(2))^6 ((6 · ()/(4)) + i (6 · ()/(4))) b^6 = 2^3 (((3)/(2)) + i ((3)/(2))) b^6 = 8 (0 + i · (-1)) b^6 = -8i Step 4: Apply De Moivre's Theorem to calculate c^6. For c^6: c^6 = (sqrt(2))^6 ((6 · ()/(3)) + i (6 · ()/(3))) c^6 = 2^3 ((2) + i (2)) c^6 = 8 (1 + i · 0) c^6 = 8 Send me the next one 📸