How do you divide radicals with variables?

Hint: We are given a radical expression. We have to simplify the expression. First, write the radical expression as a quotient of two radical expressions. Then, factorize the expression at the numerator and denominator. Then, cancel out the common factors. Then, write the result in simplified form.

Complete step-by-step solution:
First, write the radical expression involving variables.
$ \Rightarrow \sqrt[3]{{\dfrac{{12{m^2}}}{{5{n^2}}}}}$
Now, the expression involves the variables m and n. The number written over the radical symbol represents that it is the cube root of the expression.
Now, write the expression as a quotient of two radical expressions.
$ \Rightarrow \sqrt[3]{{\dfrac{{12{m^2}}}{{5{n^2}}}}} = \dfrac{{\sqrt[3]{{12{m^2}}}}}{{\sqrt[3]{{5{n^2}}}}}$
Now, rationalize the denominator to make it a perfect cube so that the radical sign must be removed from the denominator.
$ \Rightarrow \sqrt[3]{{\dfrac{{12{m^2}}}{{5{n^2}}}}} = \dfrac{{\sqrt[3]{{12{m^2}}}}}{{\sqrt[3]{{5{n^2}}}}} \times \dfrac{{\sqrt[3]{{25n}}}}{{\sqrt[3]{{25n}}}}$
Now, we will multiply the terms at the numerator and denominator.
$ \Rightarrow \sqrt[3]{{\dfrac{{12{m^2}}}{{5{n^2}}}}} = \dfrac{{\sqrt[3]{{12{m^2} \times 25n}}}}{{\sqrt[3]{{5{n^2} \times 25n}}}}$
On simplifying the expression, we get:
$ \Rightarrow \sqrt[3]{{\dfrac{{12{m^2}}}{{5{n^2}}}}} = \dfrac{{\sqrt[3]{{300{m^2}n}}}}{{\sqrt[3]{{125{n^3}}}}}$
Now, the denominator is a perfect cube, then determine the cube root of the denominator.
$ \Rightarrow \sqrt[3]{{\dfrac{{12{m^2}}}{{5{n^2}}}}} = \dfrac{{\sqrt[3]{{300{m^2}n}}}}{{5n}}$
Since the numerator is not a perfect cube, which means the numerator cannot be simplified further.

Note: The students must note that to divide the fractional radicals, the quotient rule is applied if the index value of numerator and denominator is the same. The radical symbol without number represents the square root of the expression, the radical symbol with a number such as 3 represents cube root and 4 represents fourth root, etc. The square root of the perfect square and the cube root of the perfect cube of the expression are known as radical rationales. Also, when the radicals cannot be simplified further then it is called irrational radicals.