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

Hint: The representation for multiplication of a number \[a\] with itself is \[a \times a = {a^2}\] and representation of a multiplication of a number \[a\] with itself and again multiply the result with the number \[a\] is \[a \times a \times a = {a^3}\].

Complete step-by-step solution:
The given algebraic expression is \[{\left( {3{x^3}{y^3}} \right)^2}\].
First simplify the bracket of the given expression.
As the bracket has a power of two, therefore expression can be expanded as,
\[3{x^3}{y^3} \times 3{x^3}{y^3}\]
Multiplication is a commutative operation, it means \[a \times b = b \times a\].
Therefore, rearrange the terms of the expression as shown below.
\[ \Rightarrow 3 \times 3 \times {x^3} \times {x^3} \times {y^3} \times {y^3}\]
Simplify the expression as shown below.
\[ \Rightarrow 9 \times {x^3} \times {x^3} \times {y^3} \times {y^3}\]
Now use the algebraic identities \[{a^m} \times {a^n} = {a^{m + n}}\] as shown below.
\[ \Rightarrow 9 \times {x^{3 + 3}} \times {y^{3 + 3}}\]
Simplify further as shown below.
\[ \Rightarrow 9 \times {x^6} \times {y^6}\]
Also we can omit the multiplication sign in algebraic expression as shown below.
\[ \Rightarrow 9{x^6}{y^6}\]
Thus, the simplification of the given algebraic expression is \[9{x^6}{y^6}\].

Note: Zero to the power zero is not defined in the algebraic domain. Also one to the power any number is always one. There are few more rule for solving the algebraic expression which as shown as:
The multiplication of a number \[a\] to the power \[m\] into a number \[a\] to the power \[n\] is equal to a number \[a\] to the power of sum of \[m\] and \[n\].
\[{a^m} \times {a^n} = {a^{m + n}}\]
The multiplication of a number \[a\] to the power \[m\] into a number \[b\] to the power \[m\] is equal to a number \[ab\] to the power of \[m\].
\[{a^m} \times {b^m} = {\left( {ab} \right)^m}\]
Similarly the division of a number \[a\] to the power \[m\] by a number \[a\] to the power \[n\] is equal to a number \[a\] to the power of difference of \[n\] and \[m\].
\[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\]
The division of a number \[a\] to the power \[m\] by a number \[b\] to the power \[m\] is equal to a number \[\dfrac{a}{b}\] to the power of \[m\].
\[\dfrac{{{a^m}}}{{{b^m}}} = {\left( {\dfrac{a}{b}} \right)^m}\]