Answer
$-i$
Work Step by Step
Note that
$(-i) = -1(i)$
Thus, the given expression is equivalent to:
$=[-1(i)]^9$
Use the rule $(ab)^m = a^mb^m$ to obtain:
$=(-1)^9(i)^9
\\=(-1)(i^9)
\\=-i^9
\\=-i^{8+1}$
RECALL:
(1) $a^{mn} = (a^m)^n$
(2) $i^2=-1$
(3) $a^{m+n} = a^m \cdot a^n$
Use rule (3) above to obtain:
$-i^{8+1}=-(i^8 \cdot i)$
Use rule (1) above to obtain:
$=-[(i^2)^4 \cdot i]$
Use rule (2) above to obtain:
$\\=-[(-1)^4 \cdot i]
\\=-(1 \cdot i)
\\=-(i)
\\=-i$
