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Question

Answer

$(2 - i)^{4} = - 24 i - 7$

Explanation

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$(2 - i)^{4} \overset{◯}{1} \overset{◯}{2} \overset{◯}{3} \overset{◯}{4} i^{4} - 8 i^{3} + 24 i^{2} - 32 i + 16 \overset{◯}{5} \overset{◯}{6} \overset{◯}{7} 1 + 8 i - 24 - 32 i + 16 \overset{◯}{8} - 24 i - 7$
①	$(2 - i)^{4} = (2 - i)^{2} \cdot (2 - i)^{2}$
②	Find $(2 - i)^{2}$ using formula. $(A - B)^{2} = A^{2} - 2 \cdot A \cdot B + B^{2}$ where $A = 2$ and $B = i$ . $(2 - i)^{2} = 2^{2} - 2 \cdot 2 \cdot i + i^{2} = 4 - 4 i + i^{2}$
③	Multiply each term of $(4 - 4 i + i^{2})$ by each term in $(4 - 4 i + i^{2})$ . $(4 - 4 i + i^{2}) \cdot (4 - 4 i + i^{2}) = 16 - 16 i + 4 i^{2} - 16 i + 16 i^{2} - 4 i^{3} + 4 i^{2} - 4 i^{3} + i^{4}$
④	Combine like terms: $16 - 16 i + 4 i^{2} - 16 i + 16 i^{2} - 4 i^{3} + 4 i^{2} - 4 i^{3} + i^{4} = = i^{4} - 8 i^{3} + 24 i^{2} - 32 i + 16$
⑤	$i^{4} = i^{2} \cdot i^{2} = (- 1) \cdot (- 1) = 1$
⑥	$- 8 i^{3} = - 8 \cdot i^{2} \cdot i = - 8 \cdot (- 1) \cdot i = 8 \cdot i$
⑦	$24 i^{2} = 24 \cdot (- 1) = - 24$
⑧	Combine like terms: $8 i - 32 i - 24 + 1 + 16 = - 24 i - 7$

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