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
Here are the simplified forms of the complex numbers: a) Simplify (4)/(i) Step 1: Multiply the numerator and denominator by i to eliminate i from the denominator. (4)/(i) = (4)/(i) × (i)/(i) Step 2: Perform the multiplication and use the identity i^2 = -1. (4i)/(i^2) = (4i)/(-1) = -4i The simplified form is -4i. b) Simplify (1-i)/(1+i) Step 1: Multiply the numerator and denominator by the conjugate of the denominator, which is 1-i. (1-i)/(1+i) = (1-i)/(1+i) × (1-i)/(1-i) Step 2: Expand the numerator and the denominator. Numerator: (1-i)(1-i) = 1 - i - i + i^2 = 1 - 2i + (-1) = -2i Denominator: (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 Step 3: Combine the simplified numerator and denominator. (-2i)/(2) = -i The simplified form is -i. c) Simplify (4+5i)/(6-5i) Step 1: Multiply the numerator and denominator by the conjugate of the denominator, which is 6+5i. (4+5i)/(6-5i) = (4+5i)/(6-5i) × (6+5i)/(6+5i) Step 2: Expand the numerator and the denominator. Numerator: (4+5i)(6+5i) = 4(6) + 4(5i) + 5i(6) + 5i(5i) = 24 + 20i + 30i + 25i^2 = 24 + 50i + 25(-1) = 24 + 50i - 25 = -1 + 50i Denominator: (6-5i)(6+5i) = 6^2 - (5i)^2 = 36 - 25i^2 = 36 - 25(-1) = 36 + 25 = 61 Step 3: Combine the simplified numerator and denominator. (-1+50i)/(61) = -(1)/(61) + (50)/(61)i The simplified form is -(1)/(61) + (50)/(61)i. d) Simplify (4i)/((1+2i)^2) Step 1: Expand the denominator (1+2i)^2. (1+2i)^2 = 1^2 + 2(1)(2i) + (2i)^2 = 1 + 4i + 4i^2 = 1 + 4i + 4(-1) = 1 + 4i - 4 = -3 + 4i Step 2: Rewrite the expression with the expanded denominator. (4i)/(-3+4i) Step 3: Multiply the numerator and denominator by the conjugate of the new denominator, which is -3-4i. (4i)/(-3+4i) × (-3-4i)/(-3-4i) Step 4: Expand the numerator and the denominator. Numerator: 4i(-3-4i) = 4i(-3) + 4i(-4i) = -12i - 16i^2 = -12i - 16(-1) = 16 - 12i Denominator: (-3+4i)(-3-4i) = (-3)^2 - (4i)^2 = 9 - 16i^2 = 9 - 16(-1) = 9 + 16 = 25 Step 5: Combine the simplified numerator and denominator. (16-12i)/(25) = (16)/(25) - (12)/(25)i The simplified form is (16)/(25) - (12)/(25)i. What's next?