Answer
$\dfrac{11\sqrt{2}}{8}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Simplify each term in the given expression, $
\dfrac{1}{\sqrt{2}}+\dfrac{3}{\sqrt{8}}+\dfrac{1}{\sqrt{32}}
.$ Then make the terms similar (same denominator) to combine the numerators.
$\bf{\text{Solution Details:}}$
Simplifying each term of the expression above results to
\begin{array}{l}\require{cancel}
\dfrac{1}{\sqrt{2}}\cdot\dfrac{\sqrt{2}}{\sqrt{2}}+\dfrac{3}{\sqrt{8}}\cdot\dfrac{\sqrt{2}}{\sqrt{2}}+\dfrac{1}{\sqrt{32}}\cdot\dfrac{\sqrt{2}}{\sqrt{2}}
\\\\=
\dfrac{\sqrt{2}}{\sqrt{4}}+\dfrac{3\sqrt{2}}{\sqrt{16}}+\dfrac{\sqrt{2}}{\sqrt{64}}
\\\\=
\dfrac{\sqrt{2}}{2}+\dfrac{3\sqrt{2}}{4}+\dfrac{\sqrt{2}}{8}
.\end{array}
To simplify the expression above, make the terms similar by multiplying the necessary term/s to an expression equal to $1$. Hence, the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt{2}}{2}\cdot\dfrac{4}{4}+\dfrac{3\sqrt{2}}{4}\cdot\dfrac{2}{2}+\dfrac{\sqrt{2}}{8}
\\\\=
\dfrac{4\sqrt{2}}{8}+\dfrac{6\sqrt{2}}{8}+\dfrac{\sqrt{2}}{8}
.\end{array}
To combine similar terms, add/subtract the numerators and copy the similar denominator. Hence, the expression above is equivalent to \begin{array}{l}\require{cancel}
\dfrac{4\sqrt{2}+6\sqrt{2}+\sqrt{2}}{8}
\\\\=
\dfrac{(4+6+1)\sqrt{2}}{8}
\\\\=
\dfrac{11\sqrt{2}}{8}
.\end{array}
