How do you simplify the radical expression : $\dfrac{{\sqrt {120} }}{{\sqrt {50} }}$?

Hint: This problem deals with simplifying the given radical expression. This is done by simplifying the radical expressions using the product and quotient rule for radicals. Use formulas involving radicals. Evaluate given square root and cube root functions. An algebraic expression that contains radicals is called a radical expression. We use the product and quotient rules to simplify them.

Complete step-by-step solution:
Given the radical expression $\dfrac{{\sqrt {120} }}{{\sqrt {50} }}$, consider this expression as given below:
$ \Rightarrow \dfrac{{\sqrt {120} }}{{\sqrt {50} }}$
Now multiply and divide the above expression with $\sqrt {50} $, as shown below:
$ \Rightarrow \dfrac{{\sqrt {120} }}{{\sqrt {50} }} \times \dfrac{{\sqrt {50} }}{{\sqrt {50} }}$
Now multiplying the $\sqrt {50} $ in the both of the numerator and the denominator of the above expression, as shown below:
\[ \Rightarrow \dfrac{{\sqrt {120} \times \sqrt {50} }}{{\sqrt {50} \times \sqrt {50} }}\]
We know that \[\sqrt {50} \times \sqrt {50} = {\left( {\sqrt {50} } \right)^2}\], which is further equal to \[{\left( {\sqrt {50} } \right)^2} = 50\], hence substituting, as :
\[ \Rightarrow \dfrac{{\sqrt {120 \times 50} }}{{50}} = \dfrac{{\sqrt {6000} }}{{50}}\]
Now multiplying the \[\sqrt {120} \times \sqrt {50} \] in the numerator, as shown below:
\[ \Rightarrow \dfrac{{\sqrt {6000} }}{{50}} = \dfrac{{10 \times 2\sqrt {3 \times 5} }}{{50}}\]
Now the number inside the numerator inside the root is 6000 which is factored into perfect squares inside, as the square root can be removed for the perfect squares.
\[ \Rightarrow \dfrac{{20\sqrt {15} }}{{50}} = \dfrac{{2\sqrt {15} }}{5}\]
$\therefore \dfrac{{\sqrt {120} }}{{\sqrt {50} }} = \dfrac{{2\sqrt {15} }}{5}$
The simplification of the radical expression of $\dfrac{{\sqrt {120} }}{{\sqrt {50} }}$ is equal to $\dfrac{{2\sqrt {15} }}{5}$.

The simplification of the radical expression of $\dfrac{{\sqrt {120} }}{{\sqrt {50} }}$ is equal to $\dfrac{{2\sqrt {15} }}{5}$.

Note: Please note that if you want to multiply, first coefficients are multiplied with each other and the sub-radical amounts each other, placing the latter product under the radical sign common and the result is simplified.
But if you want to divide, then the coefficients are divided among themselves and sub-radical amounts each other, placing the latter quotient under the radical common and the result is simplified.