$(1 + i)^{4} = - 4$

Explanation

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

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(1+i)4=−4

$(1 + i)^{4} = - 4$