How do you simplify \[{{\left( 2{{m}^{2}}n \right)}^{2}}.3mn?\]

Hint: Before starting evaluating any expression first we need to simplify it. Then all the calculations get much easier there. Here in the given expression first remove the parentheses by using multiplying factors. Then in the terms of the exponents we need to use the various laws for simplifying the given expression, then we have to combine all the constants. By using all the conditions given above we have to simplify the given expression.

Complete step by step solution:
Given expression is \[{{\left[ 2{{m}^{2}}n \right]}^{2}}.3mn.\]
Now we have to apply the multiplication distributive property we get, \[{{\left( xy \right)}^{a}}={{x}^{a}},{{y}^{a}}\]
The expression becomes,
\[{{2}^{2}}{{\left[ {{m}^{2}} \right]}^{2}}{{n}^{2}}.3mn\]
Solve the \[{{2}^{2}}\]we get \[4\]
\[\therefore \,\,\,4{{\left( {{m}^{2}} \right)}^{2}}{{n}^{2}},3mn\]
After that we have to apply the power rule of the exponents i.e.
\[{{\left( {{x}^{a}} \right)}^{b}}={{a}^{a.b}}\]
\[\therefore \] The expression becomes
\[\therefore 4\left( {{m}^{2.2}} \right){{n}^{2}}.3mn\]
Simplify the above expression we get,
\[4{{m}^{4}}{{n}^{2}}.3mn\]
Now, multiply the constants we get,
\[4r3{{m}^{4}}{{n}^{2}}mn\]
Simplify it we get
\[12{{m}^{4}}{{n}^{2}}mn\]
Then, we have to apply the product rule of the exponents i.e.
\[{{x}^{a}}{{x}^{b}}={{x}^{a+b}}\]
Simplify it \[12{{m}^{4+1}}{{n}^{2+1}}\]
\[\therefore \,\,\,\,12{{m}^{5}}.{{n}^{3}}\]

After simplifying the expression \[{{\left( 2{{m}^{2}}n \right)}^{2}},3mn\] we get\[,12{{m}^{5}}.{{n}^{3}}\]

Note: While simplifying the expression you need to remember some important points they are, remove the parentheses or any grouping symbol. And also use the rules of exponents for exponents. Also combine the like terms and constants.