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.