Answer
$\dfrac{3x+1}{2x^{\frac{1}{2}}}$
Work Step by Step
The LCD is $2x^{\frac{1}{2}}$.
Make the expressions similar using their LCD to obtain:
$\dfrac{1+x}{2x^{\frac{1}{2}}}+x^{\frac{1}{2}}=\dfrac{1+x}{2x^{\frac{1}{2}}}+x^{\frac{1}{2}}\cdot \dfrac{2x^{\frac{1}{2}}}{2x^{\frac{1}{2}}}$
Use the rule $a^m \cdot a^n = a^{m+n}$, then simplify to obtain:
$=\dfrac{1+x}{2x^{\frac{1}{2}}}+\dfrac{2x^{\frac{1}{2}+{\frac{1}{2}}}}{2x^{\frac{1}{2}}}$
$=\dfrac{1+x}{2x^{\frac{1}{2}}}+\dfrac{2x}{2x^{\frac{1}{2}}}$
Add the expressions together to obtain:
$=\dfrac{1+x+2x}{2x^{\frac{1}{2}}}$
$=\dfrac{3x+1}{2x^{\frac{1}{2}}}$
Hence, the correct answer is $\frac{3x+1}{2x^{\frac{1}{2}}}$.
