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
You're on a roll — here are the solutions for the remaining questions. 1. Simplify: a) (5+j4)(3+j7)(2-j3) Step 1: Multiply the first two complex numbers. (5+j4)(3+j7) = 5(3) + 5(j7) + j4(3) + j4(j7) = 15 + j35 + j12 + j^228 Since j^2 = -1: = 15 + j47 - 28 = -13 + j47 Step 2: Multiply the result from Step 1 by the third complex number. (-13 + j47)(2-j3) = -13(2) - 13(-j3) + j47(2) + j47(-j3) = -26 + j39 + j94 - j^2141 Since j^2 = -1: = -26 + j133 + 141 = (141 - 26) + j133 = 115 + j133 b) ((2-j3)(3+j2))/((4-j3)) Step 1: Simplify the numerator by multiplying the two complex numbers. (2-j3)(3+j2) = 2(3) + 2(j2) - j3(3) - j3(j2) = 6 + j4 - j9 - j^26 Since j^2 = -1: = 6 - j5 + 6 = 12 - j5 Step 2: Divide the result from Step 1 by the denominator. Multiply the numerator and denominator by the conjugate of the denominator, which is (4+j3). (12-j5)/(4-j3) = ((12-j5)(4+j3))/((4-j3)(4+j3)) Step 3: Expand the numerator and the denominator. Numerator: (12-j5)(4+j3) = 12(4) + 12(j3) - j5(4) - j5(j3) = 48 + j36 - j20 - j^215 = 48 + j16 + 15 = 63 + j16 Denominator: (4-j3)(4+j3) = 4^2 - (j3)^2 = 16 - j^29 = 16 + 9 = 25 Step 4: Combine the simplified numerator and denominator. (63+j16)/(25) = (63)/(25) + j(16)/(25) c) ( 3x + j 3x)/( x + j x) Step 1: Express the numerator and denominator in exponential form using Euler's formula, e^j = + j . 3x + j 3x = e^j3x x + j x = e^jx Step 2: Perform the division using the properties of exponents. e^j3xe^jx = e^j3x - jx = e^j(3x-x) = e^j2x Step 3: Convert the result back to trigonometric form using Euler's formula. e^j2x = 2x + j 2x All done for today! What's next? 📸