Answer
$\dfrac{\sqrt{2}}{8}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
\dfrac{2\sqrt{25}}{8\sqrt{50}}
,$ use the laws of radicals. Then rationalize the denominator.
$\bf{\text{Solution Details:}}$
Using the Quotient Rule of radicals which is given by $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}{},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{2}{8}\sqrt{\dfrac{25}{50}}
\\\\=
\dfrac{\cancel2}{\cancel24}\sqrt{\dfrac{\cancel{25}}{\cancel{25}\cdot2}}
\\\\=
\dfrac{1}{4}\sqrt{\dfrac{1}{2}}
.\end{array}
Rationalizing the numerator by multiplying the radicand by an expression equal to $1$ which will make the denominator a perfect power of the index, the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{4}\sqrt{\dfrac{1}{2}\cdot\dfrac{2}{2}}
\\\\=
\dfrac{1}{4}\sqrt{\dfrac{2}{2^2}}
.\end{array}
Using the Quotient Rule of radicals which is given by $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}{},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{4}\cdot\dfrac{\sqrt{2}}{\sqrt{2^2}}
\\\\=
\dfrac{1}{4}\cdot\dfrac{\sqrt{2}}{2}
\\\\=
\dfrac{\sqrt{2}}{8}
.\end{array}
