Answer
The required value is $\frac{\left( 2x+3 \right)}{{{\left( x+3 \right)}^{\frac{3}{5}}}}$.
Work Step by Step
Consider the expression,
$x{{\left( x+3 \right)}^{-\frac{3}{5}}}+{{\left( x+3 \right)}^{\frac{2}{5}}}$
Now, evaluate the given expression:
$\begin{align}
& x{{\left( x+3 \right)}^{-\frac{3}{5}}}+{{\left( x+3 \right)}^{\frac{2}{5}}}=x{{\left( x+3 \right)}^{-\frac{3}{5}}}+{{\left( x+3 \right)}^{\frac{5}{5}-\frac{3}{5}}} \\
& =x{{\left( x+3 \right)}^{-\frac{3}{5}}}+{{\left( x+3 \right)}^{\frac{5}{5}}}{{\left( x+3 \right)}^{-\frac{3}{5}}} \\
& =x{{\left( x+3 \right)}^{-\frac{3}{5}}}+\left( x+3 \right){{\left( x+3 \right)}^{-\frac{3}{5}}} \\
& ={{\left( x+3 \right)}^{-\frac{3}{5}}}\left( x+\left( x+3 \right) \right)
\end{align}$
Further, simplifying, we get,
$\begin{align}
& x{{\left( x+3 \right)}^{-\frac{3}{5}}}+{{\left( x+3 \right)}^{\frac{2}{5}}}={{\left( x+3 \right)}^{-\frac{3}{5}}}\left( x+x+3 \right) \\
& ={{\left( x+3 \right)}^{-\frac{3}{5}}}\left( 2x+3 \right) \\
& =\frac{\left( 2x+3 \right)}{{{\left( x+3 \right)}^{\frac{3}{5}}}}
\end{align}$
Therefore, the value of $x{{\left( x+3 \right)}^{-\frac{3}{5}}}+{{\left( x+3 \right)}^{\frac{2}{5}}}$ is $\frac{\left( 2x+3 \right)}{{{\left( x+3 \right)}^{\frac{3}{5}}}}$.
