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Write (1-2i)^(4) in simplest a+bi form. Answer:

Write (1−2i)4 (1-2 i)^{4} in simplest a+bi a+b i form.\newlineAnswer:

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Full solution

Q. Write (1−2i)4 (1-2 i)^{4} in simplest a+bi a+b i form.\newlineAnswer:
  1. Recognize pattern of powers: Recognize the pattern of powers of a complex number. \newline(1−2i)1=1−2i(1-2i)^1 = 1 - 2i\newline(1−2i)2=(1−2i)(1−2i)=1−4i+4i2=1−4i−4(1-2i)^2 = (1 - 2i)(1 - 2i) = 1 - 4i + 4i^2 = 1 - 4i - 4 (since i2=−1i^2 = -1)\newline(1−2i)3=(1−2i)(1−2i)2=(1−2i)(1−4i−4)(1-2i)^3 = (1-2i)(1-2i)^2 = (1-2i)(1 - 4i - 4)\newlineWe need to calculate (1−2i)4(1-2i)^4, which is (1−2i)2∗(1−2i)2(1-2i)^2 * (1-2i)^2.
  2. Calculate (1−2i)2(1-2i)^2: Calculate (1−2i)2(1-2i)^2.(1−2i)2=1−4i+4i2=1−4i−4=−3−4i(1-2i)^2 = 1 - 4i + 4i^2 = 1 - 4i - 4 = -3 - 4i
  3. Find (1−2i)4(1-2i)^4: Use the result from Step 22 to find (1−2i)4(1-2i)^4.(1−2i)4=(−3−4i)2(1-2i)^4 = (-3 - 4i)^2
  4. Square the result: Square the result from Step 33.\newline(−3−4i)2=(−3)2−2×3×4i+(4i)2=9−24i+16i2(-3 - 4i)^2 = (-3)^2 - 2 \times 3 \times 4i + (4i)^2 = 9 - 24i + 16i^2
  5. Simplify the expression: Simplify the expression by recognizing that i2=−1i^2 = -1. \newline9−24i+16i2=9−24i−16=−7−24i9 - 24i + 16i^2 = 9 - 24i - 16 = -7 - 24i

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