Complex Numbers

This summary of algebraic operations on complex numbers will prepare you for solving quadratic equations with no solutions and the related implications for graphing quadratic and polynomial functions.

Q&A

Can we write $i^{35}$ in other helpful ways?

Can we write $i^{35}$ in other helpful ways?As we saw in Example 8, we reduced $i^{35}$ to $i^{3}$ by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of $i^{35}$ may be more useful. Table 1 shows some other possible factorizations.

Factorization of $i^{35}$	$i^{34} \cdot i$	$i^{33} \cdot i^{2}$	$i^{31} \cdot i^{4}$	$i^{19} \cdot i^{16}$
Reduced form	$\left(i^{2}\right)^{17} \cdot i$	$i^{33} \cdot(-1)$	$i^{31} \cdot 1$	$i^{19} \cdot\left(i^{4}\right)^{4}$
Simplified form	$(-1)^{17} \cdot i$	$-i^{33}$	$i^{31}$	$i^{19}$