Answer
$-i$
Work Step by Step
Note that
$(-i) = -1(i)$
Thus, the given expression is equivalent to:
$=[-1(i)]^{13}$
Use the rule $(ab)^m = a^mb^m$ to obtain:
$=(-1)^{13}(i)^{13}
\\=(-1)(i^{13})
\\=-i^{13}
\\=-i^{12+1}$
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:
$-i^{12+1}=-(i^{12} \cdot i)$
Use rule (1) above to obtain:
$=-[(i^2)^6 \cdot i]$
Use rule (2) above to obtain:
$\\=-[(-1)^6 \cdot i]
\\=-(1 \cdot i)
\\=-(i)
\\=-i$
