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's the solution to problem 8: Given complex numbers: Z_1 = 2 + 3i Z_2 = 2 - i a) Express (Z_1)/(Z_2) in the form a + bi where a, b R. Step 1: Write the expression for (Z_1)/(Z_2). (Z_1)/(Z_2) = (2 + 3i)/(2 - i) Step 2: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 2 - i is 2 + i. (Z_1)/(Z_2) = (2 + 3i)/(2 - i) × (2 + i)/(2 + i) Step 3: Perform the multiplication in the numerator. (2 + 3i)(2 + i) = 2(2) + 2(i) + 3i(2) + 3i(i) = 4 + 2i + 6i + 3i^2 Since i^2 = -1: = 4 + 8i - 3 = 1 + 8i Step 4: Perform the multiplication in the denominator. (2 - i)(2 + i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 Step 5: Combine the numerator and denominator to get the result in a+bi form. (Z_1)/(Z_2) = (1 + 8i)/(5) = (1)/(5) + (8)/(5)i The expression in the form a+bi is (1)/(5) + (8)/(5)i. b) Find the modulus and argument of Z_1 × Z_2. Step 1: Calculate the product Z_1 × Z_2. Z_1 × Z_2 = (2 + 3i)(2 - i) = 2(2) + 2(-i) + 3i(2) + 3i(-i) = 4 - 2i + 6i - 3i^2 Since i^2 = -1: = 4 + 4i - 3(-1) = 4 + 4i + 3 = 7 + 4i Step 2: Find the modulus of Z_1 × Z_2. For a complex number z = x + yi, the modulus is |z| = sqrt(x^2 + y^2). |Z_1 × Z_2| = |7 + 4i| = sqrt(7^2 + 4^2) = sqrt(49 + 16) = sqrt(65) Step 3: Find the argument of Z_1 × Z_2. For z = x + yi in the first quadrant (x>0, y>0), the argument is (z) = ((y)/(x)). Since 7 + 4i is in the first quadrant: (Z_1 × Z_2) = ((4)/(7)) The modulus is sqrt(65) and the argument is ((4)/(7)). c) Given also that Z_3 is a complex number such that |Z_3| = 5 and the argument of Z_3 is ()/(3), express (Z_1 × Z_2)/(Z_3) in polar form. Step 1: Write Z_3 in polar form. Given |Z_3| = 5 and (Z_3) = ()/(3). Z_3 = 5 ((()/(3)) + i (()/(3))) Step 2: Write Z_1 × Z_2 in polar form using the results from part b). Let Z_12 = Z_1 × Z_2. |Z_12| = sqrt(65) and (Z_12) = ((4)/(7)). Z_12 = sqrt(65) ((((4)/(7))) + i (((4)/(7)))) Step 3: Express Z_12Z_3 in polar form. If z_a = r_a( _a + i _a) and z_b = r_b( _b + i _b), then (z_a)/(z_b) = (r_a)/(r_b)((_a - _b) + i (_a - _b)). Here, r_a = sqrt(65), _a = ((4)/(7)), r_b = 5, _b = ()/(3). (Z_1 × Z_2)/(Z_3) = sqrt(65)5 ((((4)/(7)) - ()/(3)) + i (((4)/(7)) - ()/(3))) The expression in polar form is sqrt(65)5 ((((4)/(7)) - ()/(3)) + i (((4)/(7)) - ()/(3))).