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 \((-1)^{17} \cdot i\) \(-i^{33}\) \(i^{31}\) \(i^{19}\)

Table 1

Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.