Use Power Reduction Rule: \({i}^{n}={i}^{n\text{ mod }4}\).
\[4\times 1-3{\imath }^{9}+33{\imath }^{11}-4{\imath }^{10}-2-\]
Use Power Reduction Rule: \({i}^{n}={i}^{n\text{ mod }4}\).
\[4\times 1-3\imath +33{\imath }^{11}-4{\imath }^{10}-2-\]
Use Power Reduction Rule: \({i}^{n}={i}^{n\text{ mod }4}\).
\[4\times 1-3\imath +33{\imath }^{3}-4{\imath }^{10}-2-\]
Isolate \({\imath }^{2}\).
\[4\times 1-3\imath +33{\imath }^{2}\imath -4{\imath }^{10}-2-\]
Use Square Rule: \({i}^{2}=-1\).
\[4\times 1-3\imath +33\times -1\times \imath -4{\imath }^{10}-2-\]
Use Power Reduction Rule: \({i}^{n}={i}^{n\text{ mod }4}\).
\[4\times 1-3\imath +33\times -1\times \imath -4{\imath }^{2}-2-\]
Use Square Rule: \({i}^{2}=-1\).
\[4\times 1-3\imath +33\times -1\times \imath -4\times -1-2-\]
Simplify \(4\times 1\) to \(4\).
\[4-3\imath +33\times -1\times \imath -4\times -1-2-\]
Simplify \(33\times -1\times \imath \) to \(-33\imath \).
\[4-3\imath -33\imath -4\times -1-2-\]
Simplify \(4\times -1\) to \(-4\).
\[4-3\imath -33\imath -(-4)-2-\]
Remove parentheses.
\[4-3\imath -33\imath +4-2-\]
Collect like terms.
\[(4+4-2)+(-3\imath -33\imath )-1\]
Simplify.
\[6-36\imath -1\]
Collect like terms.
\[(6-1)-36\imath \]
Simplify.
\[5-36\imath \]
5-36*IM
