Answer
The error in the expression is $\sqrt{-9}+\sqrt{-16}\ne \sqrt{-25}$ but the expression $\sqrt{-25}=i\sqrt{25}=5i$ is correct.
Work Step by Step
Consider the expression, $\sqrt{-9}+\sqrt{-16}=\sqrt{-25}=i\sqrt{25}=5i$
Take the first two imaginary parts.
$\sqrt{-9}+\sqrt{-16}$
The principal square root of a negative number is such that for any positive real number $b$,
$\sqrt{-b}=i\sqrt{b}$
Express all the square roots of negative numbers in terms of $i$.
$\begin{align}
& \sqrt{-9}+\sqrt{-16}=i\sqrt{9}+i\sqrt{16} \\
& =i\sqrt{{{3}^{2}}}+i\sqrt{{{4}^{2}}} \\
& =3i+4i \\
& =7i
\end{align}$
The expression $\sqrt{-9}+\sqrt{-16}$ is equal to $7i$.
The error in the expression is such that, $\sqrt{-9}+\sqrt{-16}\ne \sqrt{-25}$ and $\sqrt{-25}=i\sqrt{25}=5i$
The number $\sqrt{-9}$ and $-16$ cannot be added directly.
Therefore, the error in the expression is $\sqrt{-9}+\sqrt{-16}\ne \sqrt{-25}$, but the expression$\sqrt{-25}=i\sqrt{25}=5i$ is correct.
