Answer
$11-5i$
Work Step by Step
Since $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b},$ the given expression, $
(8-\sqrt{-1})-(-3+\sqrt{-16})
,$ is equivalent to
\begin{align*}
&
(8-\sqrt{-1})-(-3+\sqrt{-1}\cdot\sqrt{16})
\\&=
(8-\sqrt{-1})-(-3+\sqrt{-1}\cdot4)
\\&=
(8-\sqrt{-1})-(-3+4\sqrt{-1})
.\end{align*}
Since $i=\sqrt{-1},$ the expression above is equivalent to
\begin{align*}
(8-i)-(-3+4i)
.\end{align*}
Removing the grouping symbols, the expression above is equivalent to
\begin{align*}
8-i+3-4i
.\end{align*}
Combining the real parts and the imaginary parts, the expression above is equivalent to
\begin{align*}
&
(8+3)+(-i-4i)
\\&=
11+(-5i)
\\&=
11-5i
.\end{align*}
Hence, the simplified form of the given expression is $
11-5i
$.
