Answer
$2i\sqrt{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\sqrt{-8}
,$ use the properties of radicals and the definition of an imaginary number.
$\bf{\text{Solution Details:}}$
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to\begin{array}{l}\require{cancel}
\sqrt{-1}\cdot\sqrt{8}
.\end{array}
Using the definition of an imaginary number which is given by $i=\sqrt{-1},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
i\cdot\sqrt{8}
.\end{array}
Simplifying the radical by extracting the root of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
i\cdot\sqrt{4\cdot2}
\\\\=
i\cdot\sqrt{(2)^2\cdot2}
\\\\=
i\cdot2\sqrt{2}
\\\\=
2i\sqrt{2}
.\end{array}
