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.
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Here are the solutions to the problems. Problem 1: Simplify (1-i)^1364 Step 1: Rewrite the numerator. (1-i)^1364 = (1-i)^12(1-i)64 Step 2: Express (1-i)^12 as a power of (1-i)^2. (((1-i)^2)^6(1-i))/(64) Step 3: Calculate (1-i)^2. (1-i)^2 = 1^2 - 2(1)(i) + i^2 = 1 - 2i - 1 = -2i Step 4: Substitute this back into the expression. ((-2i)^6(1-i))/(64) Step 5: Calculate (-2i)^6. (-2i)^6 = (-2)^6 i^6 = 64 i^6 Step 6: Simplify i^6. i^6 = i^4 · i^2 = 1 · (-1) = -1 Step 7: Substitute i^6 = -1 back into the expression. (64(-1)(1-i))/(64) = (-64(1-i))/(64) Step 8: Cancel out the 64 in the numerator and denominator. -(1-i) Step 9: Distribute the negative sign. -1+i The simplified form is -1+i. --- Problem 2: Verify ((C_1)/(C_2)) = C_1C_2 given C_1 = 7-4i and C_2 = 2-3i. Part A: Calculate the Left Hand Side (LHS) LHS: ((C_1)/(C_2)) Step 1: Calculate (C_1)/(C_2). (C_1)/(C_2) = (7-4i)/(2-3i) Step 2: Multiply the numerator and denominator by the conjugate of the denominator, which is 2+3i. (7-4i)/(2-3i) × (2+3i)/(2+3i) = ((7-4i)(2+3i))/((2-3i)(2+3i)) Step 3: Expand the numerator and denominator. Numerator: (7-4i)(2+3i) = 7(2) + 7(3i) - 4i(2) - 4i(3i) = 14 + 21i - 8i - 12i^2 = 14 + 13i - 12(-1) = 14 + 13i + 12 = 26 + 13i. Denominator: (2-3i)(2+3i) = 2^2 - (3i)^2 = 4 - 9i^2 = 4 - 9(-1) = 4 + 9 = 13. Step 4: Simplify the fraction. (26+13i)/(13) = (26)/(13) + (13i)/(13) = 2+i Step 5: Find the conjugate of the result. ((C_1)/(C_2)) = (2+i) = 2-i So, LHS = 2-i. Part B: Calculate the Right Hand Side (RHS) RHS: C_1C_2 Step 1: Find the conjugates of C_1 and C_2. C_1 = 7-4i = 7+4i C_2 = 2-3i = 2+3i Step 2: Calculate C_1C_2. C_1C_2 = (7+4i)/(2+3i) Step 3: Multiply the numerator and denominator by the conjugate of the denominator, which is 2-3i. (7+4i)/(2+3i) × (2-3i)/(2-3i) = ((7+4i)(2-3i))/((2+3i)(2-3i)) Step 4: Expand the numerator and denominator. Numerator: (7+4i)(2-3i) = 7(2) - 7(3i) + 4i(2) - 4i(3i) = 14 - 21i + 8i - 12i^2 = 14 - 13i - 12(-1) = 14 - 13i + 12 = 26 - 13i. Denominator: (2+3i)(2-3i) = 2^2 - (3i)^2 = 4 - 9i^2 = 4 - 9(-1) = 4 + 9 = 13. Step 5: Simplify the fraction. (26-13i)/(13) = (26)/(13) - (13i)/(13) = 2-i So, RHS = 2-i. Conclusion: Since LHS = 2-i and RHS = 2-i, we have verified that ((C_1)/(C_2)) = C_1C_2. Send me the next one 📸