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

Recognize pattern of powers: Recognize the pattern of powers of a complex number. (1-2i)^1 = 1 - 2i (1-2i)^2 = (1 - 2i)(1 - 2i) = 1 - 4i + 4i^2 = 1 - 4i - 4 (since i^2 = -1) (1-2i)^3 = (1-2i)(1-2i)^2 = (1-2i)(1 - 4i - 4) We need to calculate (1-2i)^4, which is (1-2i)^2 * (1-2i)^2.Calculate (1-2i)^2: Calculate (1-2i)^2. (1-2i)^2 = 1 - 4i + 4i^2 = 1 - 4i - 4 = -3 - 4iFind (1-2i)^4: Use the result from Step 2 to find (1-2i)^4. (1-2i)^4 = (-3 - 4i)^2Square the result: Square the result from Step 3. (-3 - 4i)^2 = (-3)^2 - 2 * 3 * 4i + (4i)^2 = 9 - 24i + 16i^2Simplify the expression: Simplify the expression by recognizing that i^2 = -1. 9 - 24i + 16i^2 = 9 - 24i - 16 = -7 - 24i

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

Write $(1-2 i)^{4}$ in simplest $a+b i$ form. $\newline$ Answer:

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Q. Write

(1-2 i)^{4}

in simplest

a+b i

form.

\newline

Answer:

Recognize pattern of powers: Recognize the pattern of powers of a complex number. $\newline$ $(1-2i)^1 = 1 - 2i$ $\newline$ $(1-2i)^2 = (1 - 2i)(1 - 2i) = 1 - 4i + 4i^2 = 1 - 4i - 4$ (since $i^2 = -1$ ) $\newline$ $(1-2i)^3 = (1-2i)(1-2i)^2 = (1-2i)(1 - 4i - 4)$ $\newline$ We need to calculate $(1-2i)^4$ , which is $(1-2i)^2 * (1-2i)^2$ .
Calculate $(1-2i)^2$ : Calculate $(1-2i)^2$ . $(1-2i)^2 = 1 - 4i + 4i^2 = 1 - 4i - 4 = -3 - 4i$
Find $(1-2i)^4$ : Use the result from Step $2$ to find $(1-2i)^4$ . $(1-2i)^4 = (-3 - 4i)^2$
Square the result: Square the result from Step $3$ . $\newline$ $(-3 - 4i)^2 = (-3)^2 - 2 \times 3 \times 4i + (4i)^2 = 9 - 24i + 16i^2$
Simplify the expression: Simplify the expression by recognizing that $i^2 = -1$ . $\newline$ $9 - 24i + 16i^2 = 9 - 24i - 16 = -7 - 24i$