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
Step 1: Expand the denominator (1-i)^3. We use the binomial expansion formula (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. Here, a=1 and b=i. (1-i)^3 = 1^3 - 3(1)^2(i) + 3(1)(i)^2 - i^3 Recall that i^2 = -1 and i^3 = i^2 · i = (-1)i = -i. (1-i)^3 = 1 - 3i + 3(-1) - (-i) (1-i)^3 = 1 - 3i - 3 + i Combine the real and imaginary parts: (1-i)^3 = (1-3) + (-3i+i) (1-i)^3 = -2 - 2i Step 2: Substitute the expanded denominator back into the expression. The expression becomes: (4+3i)/(-2-2i) Step 3: Rationalize the denominator. To rationalize, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of -2-2i is -2+2i. (4+3i)/(-2-2i) × (-2+2i)/(-2+2i) Step 4: Multiply the numerators. (4+3i)(-2+2i) = 4(-2) + 4(2i) + 3i(-2) + 3i(2i) = -8 + 8i - 6i + 6i^2 Substitute i^2 = -1: = -8 + 2i + 6(-1) = -8 + 2i - 6 = -14 + 2i Step 5: Multiply the denominators. This is in the form (a-b)(a+b) = a^2 - b^2. (-2-2i)(-2+2i) = (-2)^2 - (2i)^2 = 4 - (4i^2) Substitute i^2 = -1: = 4 - 4(-1) = 4 + 4 = 8 Step 6: Combine the simplified numerator and denominator. (-14+2i)/(8) Step 7: Simplify the fraction by dividing both the real and imaginary parts by 8. (-14)/(8) + (2i)/(8) -(7)/(4) + (1)/(4)i The final answer is -(7)/(4) + (1)/(4)i.