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
12. Simplify: a) (5+4i)(3+7i)(2-3i) Step 1: Multiply the first two complex numbers. (5+4i)(3+7i) = 5(3) + 5(7i) + 4i(3) + 4i(7i) = 15 + 35i + 12i + 28i^2 Since i^2 = -1: = 15 + 47i - 28 = -13 + 47i Step 2: Multiply the result by the third complex number. (-13+47i)(2-3i) = -13(2) - 13(-3i) + 47i(2) + 47i(-3i) = -26 + 39i + 94i - 141i^2 Since i^2 = -1: = -26 + 133i + 141 = 115 + 133i The simplified expression is 115 + 133i. b) ((2-3i)(3+2i))/((4-3i)) Step 1: Multiply the complex numbers in the numerator. (2-3i)(3+2i) = 2(3) + 2(2i) - 3i(3) - 3i(2i) = 6 + 4i - 9i - 6i^2 Since i^2 = -1: = 6 - 5i + 6 = 12 - 5i Step 2: Divide the result by the denominator. (12-5i)/(4-3i) Multiply the numerator and denominator by the conjugate of the denominator, which is (4+3i). ((12-5i)(4+3i))/((4-3i)(4+3i)) Step 3: Expand the numerator. (12-5i)(4+3i) = 12(4) + 12(3i) - 5i(4) - 5i(3i) = 48 + 36i - 20i - 15i^2 = 48 + 16i + 15 = 63 + 16i Step 4: Expand the denominator. (4-3i)(4+3i) = 4^2 - (3i)^2 = 16 - 9i^2 = 16 + 9 = 25 Step 5: Combine the numerator and denominator and express in a+bi form. (63+16i)/(25) = (63)/(25) + (16)/(25)i The simplified expression is (63)/(25) + (16)/(25)i. c) (3x+i3x)/( x+i x) Step 1: Use Euler's formula, e^i = + i. (3x+i3x)/( x+i x) = e^i3xe^ix Step 2: Apply the rules of exponents for division. e^i3x - ix = e^i(3x-x) = e^i2x Step 3: Convert back to trigonometric form using Euler's formula. e^i2x = (2x) + i(2x) The simplified expression is (2x) + i(2x). 13. Express the following in the form a+ib. a) (2+3i)/(i(4-5i)) + (2)/(i) Step 1: Simplify the first term, (2+3i)/(i(4-5i)). First, simplify the denominator: i(4-5i) = 4i - 5i^2 = 4i - 5(-1) = 5+4i. So the term becomes (2+3i)/(5+4i). Multiply the numerator and denominator by the conjugate of the denominator, (5-4i): ((2+3i)(5-4i))/((5+4i)(5-4i)) = (2(5) + 2(-4i) + 3i(5) + 3i(-4i))/(5^2 - (4i)^2) = (10 - 8i + 15i - 12i^2)/(25 - 16i^2) = (10 + 7i + 12)/(25 + 16) = (22+7i)/(41) = (22)/(41) + (7)/(41)i Step 2: Simplify the second term, (2)/(i). Multiply the numerator and denominator by -i: (2)/(i) · (-i)/(-i) = (-2i)/(-i^2) = (-2i)/(-(-1)) = (-2i)/(1) = -2i Step 3: Add the simplified terms. ((22)/(41) + (7)/(41)i) + (-2i) = (22)/(41) + ((7)/(41) - 2)i = (22)/(41) + ((7)/(41) - (82)/(41))i = (22)/(41) - (75)/(41)i The expression in the form a+ib is (22)/(41) - (75)/(41)i. b) (1)/(2+3i) + (1)/(1-2i) Step 1: Simplify the first term, (1)/(2+3i). Multiply the numerator and denominator by the conjugate of the denominator, (2-3i): (1)/(2+3i) · (2-3i)/(2-3i) = (2-3i)/(2^2 - (3i)^2) = (2-3i)/(4 - 9i^2) = (2-3i)/(4+9) = (2-3i)/(13) = (2)/(13) - (3)/(13)i Step 2: Simplify the second term, (1)/(1-2i). Multiply the numerator and denominator by the conjugate of the denominator, (1+2i): (1)/(1-2i) · (1+2i)/(1+2i) = (1+2i)/(1^2 - (2i)^2) = (1+2i)/(1 - 4i^2) = (1+2i)/(1+4) = (1+2i)/(5) = (1)/(5) + (2)/(5)i Step 3: Add the simplified terms. ((2)/(13) - (3)/(13)i) + ((1)/(5) + (2)/(5)i) Group the real and imaginary parts: ((2)/(13) + (1)/(5)) + (-(3)/(13) + (2)/(5))