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 number expression. Let the given expression be W. W = (Z-1-i)/(3+2i) Step 2: Apply the property of conjugates for division. The conjugate of a quotient is the quotient of the conjugates: ((A)/(B))^ = (A^)/(B^*). So, the conjugate of W, denoted as W^*, is: W^ = ((Z-1-i)^)/((3+2i)^*) Step 3: Find the conjugate of the numerator. The conjugate of a sum or difference is the sum or difference of the conjugates. The conjugate of a real number is itself, and the conjugate of i is -i. (Z-1-i)^ = Z^ - 1^ - i^ = Z^ - 1 - (-i) = Z^ - 1 + i Step 4: Find the conjugate of the denominator. The conjugate of a+bi is a-bi. (3+2i)^* = 3 - 2i Step 5: Substitute the conjugates back into the expression for W^*. W^ = (Z^ - 1 + i)/(3 - 2i) Step 6: Rationalize the denominator. Multiply the numerator and the denominator by the conjugate of the denominator, which is 3+2i. W^ = (Z^ - 1 + i)/(3 - 2i) × (3 + 2i)/(3 + 2i) Step 7: Multiply the denominators. (3 - 2i)(3 + 2i) = 3^2 - (2i)^2 = 9 - 4i^2 = 9 - 4(-1) = 9 + 4 = 13 Step 8: Multiply the numerators. (Z^ - 1 + i)(3 + 2i) = Z^(3) + Z^*(2i) - 1(3) - 1(2i) + i(3) + i(2i) = 3Z^ + 2iZ^ - 3 - 2i + 3i + 2i^2 = 3Z^ + 2iZ^ - 3 + i + 2(-1) = 3Z^ + 2iZ^ - 3 + i - 2 = 3Z^ + 2iZ^ - 5 + i Step 9: Combine the results to get the final conjugate expression. W^ = (3Z^ + 2iZ^* - 5 + i)/(13) The conjugate of the expression is: (3Z^ + 2iZ^ - 5 + i)/(13) Send me the next one 📸