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
To express the complex number (z_1)/(z_2) in Cartesian form, we multiply the numerator and the denominator by the conjugate of the denominator. Step 1: Identify the complex numbers z_1 and z_2. We are given z_1 = 1+i and z_2 = 2-i. Step 2: Set up the division. (z_1)/(z_2) = (1+i)/(2-i) Step 3: Multiply the numerator and denominator by the conjugate of the denominator. The denominator is 2-i. Its conjugate is 2+i. (1+i)/(2-i) × (2+i)/(2+i) Step 4: Expand the numerator. (1+i)(2+i) = 1(2) + 1(i) + i(2) + i(i) = 2 + i + 2i + i^2 Since i^2 = -1: = 2 + 3i - 1 = 1 + 3i Step 5: Expand the denominator. (2-i)(2+i) = 2^2 - (i)^2 = 4 - i^2 Since i^2 = -1: = 4 - (-1) = 4 + 1 = 5 Step 6: Combine the simplified numerator and denominator. (1+3i)/(5) Step 7: Express in Cartesian form a+bi. (1)/(5) + (3)/(5)i The Cartesian form is (1)/(5) + (3)/(5)i. That's 2 down. 3 left today — send the next one.