We will have the following:
[tex]\sqrt[]{\frac{16}{81}}=\frac{\sqrt[]{16}}{\sqrt[]{81}}=\frac{4}{9}[/tex]
So, the expression in its simplest form is 4/9.
***Explanation***
First we are given the expression:
[tex]\sqrt[]{\frac{16}{81}}[/tex]
From the properties of roots, we will have that if a root contains a fraction, then we can rewrite the expression applying the root to the numerator and denominator [Separately], this is:
[tex]\sqrt[]{\frac{16}{81}}=\frac{\sqrt[]{16}}{\sqrt[]{81}}[/tex]
Since, these expressions are equivalent, then we solve for each one, that is:
[tex]\begin{cases}\sqrt[]{16}=4 \\ \& \\ \sqrt[]{81}=9\end{cases}[/tex]
Then, we simply replace in the fraction form:
[tex]\sqrt[]{\frac{16}{81}}=\frac{4}{9}[/tex]
