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
To evaluate (1-2i)^-3, we first rewrite it as a fraction and then calculate the cube of the denominator. Step 1: Rewrite the expression. (1-2i)^-3 = (1)/((1-2i)^3) Step 2: Calculate (1-2i)^3. We can use the binomial expansion (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. Here a=1 and b=2i. (1-2i)^3 = 1^3 - 3(1)^2(2i) + 3(1)(2i)^2 - (2i)^3 = 1 - 6i + 3(4i^2) - 8i^3 Substitute i^2 = -1 and i^3 = i^2 · i = -i: = 1 - 6i + 3(4(-1)) - 8(-i) = 1 - 6i - 12 + 8i Combine the real and imaginary parts: = (1-12) + (-6+8)i = -11 + 2i Step 3: Substitute this result back into the fraction. (1)/((1-2i)^3) = (1)/(-11+2i) Step 4: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of (-11+2i) is (-11-2i). (1)/(-11+2i) × (-11-2i)/(-11-2i) Step 5: Simplify the expression. The numerator is 1 × (-11-2i) = -11-2i. The denominator is of the form (a+bi)(a-bi) = a^2+b^2. Here a=-11 and b=2. (-11)^2 + (2)^2 = 121 + 4 = 125 So the expression becomes: (-11-2i)/(125) Step 6: Separate into real and imaginary parts. -(11)/(125) - (2)/(125)i Comparing this with the given options, it matches option (a). The final answer is (a) (-(11)/(125) - (2)/(125)i). That's 2 down. 3 left today — send the next one.