How do you simplify \[{{\left( 3x{{y}^{3}} \right)}^{2}}{{\left( xy \right)}^{6}}\]?

Hint: We can simplify this expression using basic exponent formulas. First we have to remove the parentheses by applying power to the terms and making like terms as one by adding the exponents. By applying all necessary formulas needed we will get the simplest form.
Before going to solve let us know a few basic formulas of exponents and powers. They are
\[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\]
\[{{\left( \dfrac{a}{b} \right)}^{n}}=\dfrac{{{a}^{n}}}{{{b}^{n}}}\]
\[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\]
\[{{a}^{m}}.{{a}^{n}}={{a}^{m+n}}\]
\[\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\]
These are the basic formulas of exponents and we will use them in the problem where ever needed
Given equation is
\[{{\left( 3x{{y}^{3}} \right)}^{2}}{{\left( xy \right)}^{6}}\]
Now we have to use the multiplicative distributive property i.e., \[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\] to apply the power to all terms in the expression.
Using this formula our expression will become
\[\Rightarrow {{3}^{2}}{{x}^{2}}{{\left( {{y}^{3}} \right)}^{2}}{{x}^{6}}{{y}^{6}}\]
Now we have two powers to y. By applying power rules we can simplify it. i.e., \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\]
Using this power rule our expression will look like
\[\Rightarrow {{3}^{2}}{{x}^{2}}{{y}^{3.2}}{{x}^{6}}{{y}^{6}}\]
By simplifying it we will get
\[\Rightarrow {{3}^{2}}{{x}^{2}}{{y}^{6}}{{x}^{6}}{{y}^{6}}\]
Now we have to rearrange the terms in such a way that like terms come together.
\[\Rightarrow {{3}^{2}}{{x}^{2}}{{x}^{6}}{{y}^{6}}{{y}^{6}}\]
Now we know that \[{{3}^{2}}=9\] so we will get
\[\Rightarrow 9{{x}^{2}}{{x}^{6}}{{y}^{6}}{{y}^{6}}\]
Here we have two terms of each variable x and y. We have to make them as one. Now by applying an additional rule from the above formulas we can further simplify it.
By applying addition rule i.e., \[{{a}^{m}}.{{a}^{n}}={{a}^{m+n}}\] we will get
\[\Rightarrow 9{{x}^{6+2}}{{y}^{6+6}}\]
By simplifying it we will get
\[\Rightarrow 9{{x}^{8}}{{y}^{12}}\]
So by simplifying the given expression we will get \[9{{x}^{8}}{{y}^{12}}\] as the most simplified form.

Note:
If we have good knowledge on formulas of exponents we can do this in 2 steps. Otherwise we have to follow each step to think about what formula we have to apply next. we can also solve it in parenthesis and then multiplying both terms is also a method.