Answer
$8-2i$
Work Step by Step
Since $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b},$ the given expression, $
(10+\sqrt{-9})-(2+\sqrt{-25})
,$ is equivalent to
\begin{align*}
&
(10+\sqrt{-1}\cdot\sqrt{9})-(2+\sqrt{-1}\cdot\sqrt{25})
\\&=
(10+\sqrt{-1}\cdot3)-(2+\sqrt{-1}\cdot5)
\\&=
(10+3\sqrt{-1})-(2+5\sqrt{-1})
.\end{align*}
Since $i=\sqrt{-1},$ the expression above is equivalent to
\begin{align*}
(10+3i)-(2+5i)
.\end{align*}
Removing the grouping symbols, the expression above is equivalent to
\begin{align*}
10+3i-2-5i
.\end{align*}
Combining the real parts and the imaginary parts, the expression above is equivalent to
\begin{align*}
&
(10-2)+(3i-5i)
\\&=
8+(-2i)
\\&=
8-2i
.\end{align*}
Hence, the simplified form of the given expression is $
8-2i
$.
