Answer
$\color{blue}{\dfrac{11\sqrt2}{8}}$
Work Step by Step
Simplify the denominators to obtain:
$=\dfrac{1}{\sqrt2} + \dfrac{3}{\sqrt{4(2)}} + \dfrac{1}{\sqrt{16(2)}}
\\=\dfrac{1}{\sqrt2} + \dfrac{3}{2\sqrt{2}} + \dfrac{1}{4\sqrt{2}}$
Make the expressions similar by using their LCD of $4\sqrt{2}$ to obtain:
$=\dfrac{1(4)}{\sqrt2(4)} + \dfrac{3(2)}{2\sqrt2(2)}+\dfrac{1}{4\sqrt{2}}
\\=\dfrac{4}{4\sqrt2}+\dfrac{6}{4\sqrt2}+\dfrac{1}{4\sqrt2}$
Add the numerators and copy the denominator to obtain:
$=\dfrac{4+6+1}{4\sqrt2}
\\=\dfrac{11}{4\sqrt2}$
Rationalize the denominator by multiplying $\sqrt2$ to both the numerator and the denominator to obtain:
$=\dfrac{11(\sqrt2)}{4\sqrt2(\sqrt2)}
\\=\dfrac{11\sqrt2}{4(2)}
\\=\color{blue}{\dfrac{11\sqrt2}{8}}$
