Answer
The simplified form of the expression $\frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}$ is $\frac{\sqrt{x}-\sqrt{x+3}}{3\sqrt{x\left( x+3 \right)}}$.
Work Step by Step
Consider the provided expression, $\frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}$
Multiply the numerator and denominator by $\sqrt{x}\sqrt{x+3}$.
Therefore,
$\frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}=\frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}\cdot \frac{\sqrt{x}\sqrt{x+3}}{\sqrt{x}\sqrt{x+3}}$
Apply the distributive property in the numerator: $a\left( b+c \right)=ab+ac$
Therefore,
$$ $\begin{align}
& \frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}=\frac{\frac{1}{\sqrt{x+3}}\sqrt{x}\sqrt{x+3}-\frac{1}{\sqrt{x}}\sqrt{x}\sqrt{x+3}}{3\sqrt{x}\sqrt{x+3}} \\
& =\frac{\sqrt{x}-\sqrt{x+3}}{3\sqrt{x}\sqrt{x+3}} \\
& =\frac{\sqrt{x}+\sqrt{x+3}}{3\sqrt{x\left( x+3 \right)}}
\end{align}$ $$
Therefore, the simplified form of the expression $\frac{\frac{1}{\sqrt{x+3}}-\frac{1}{\sqrt{x}}}{3}$ is $\frac{\sqrt{x}-\sqrt{x+3}}{3\sqrt{x\left( x+3 \right)}}$.
