How do you simplify \[{\left( {27{x^6}{y^{12}}} \right)^{\dfrac{2}{3}}}\] ?

Hint: In this question, we will solve by breaking the brackets and transform the number into its prime factors, and then we will get the expression in the exponents for number and each variable and applying the formula \[{\left( {ab} \right)^m} = {a^m} \times {b^m}\] , and then we will apply the exponent identity \[{\left( {{a^m}} \right)^n} = {a^{m \times n}}\] , and then further simplify the expression to get the required result.

Complete step by step solution:
Exponents are defined as when an expression or a statement of specific natural numbers are represented as a repeated power by multiplication of its units then the resulting number is called as an exponent. The resulting set of numbers are the same as the original sequence.
Given expression \[{\left( {27{x^6}{y^{12}}} \right)^{\dfrac{2}{3}}}\] ,
Now firstly write \[27\] in prime factors, we get,
\[ \Rightarrow 27 = 3 \times 3 \times 3\] ,
Now here we can see that 27 can be written as 3 is multiplied 3 times, this can be written as,\[ \Rightarrow 27 = 3 \times 3 \times 3 = {3^3}\] ,
Therefore the given expression can be written as,
\[ \Rightarrow {\left( {27{x^6}{y^{12}}} \right)^{\dfrac{2}{3}}} = {\left( {{3^3} \times {x^6} \times {y^{12}}} \right)^{\dfrac{2}{3}}}\] ,
Now rewrite the expression using identity \[{\left( {ab} \right)^m} = {a^m} \times {b^m}\] , we get,
\[ \Rightarrow {\left( {27{x^6}{y^{12}}} \right)^{\dfrac{2}{3}}} = {\left( {{3^3}} \right)^{\dfrac{2}{3}}} \times {\left( {{x^6}} \right)^{\dfrac{2}{3}}} \times {\left( {{y^{12}}} \right)^{\dfrac{2}{3}}}\] ,
Now using the exponent identity, \[{\left( {{a^m}} \right)^n} = {a^{m \times n}}\] for each term we get,
\[ \Rightarrow {\left( {27{x^6}{y^{12}}} \right)^{\dfrac{2}{3}}} = \left( {{3^{3 \times \dfrac{2}{3}}}} \right) \times \left( {{x^{6 \times \dfrac{2}{3}}}} \right) \times \left( {{y^{12 \times \dfrac{2}{3}}}} \right)\] ,
Now simplifying in the powers we get,
\[ \Rightarrow {\left( {27{x^6}{y^{12}}} \right)^{\dfrac{2}{3}}} = \left( {{3^2}} \right) \times \left( {{x^{2 \times 2}}} \right) \times \left( {{y^{4 \times 2}}} \right)\] ,
Now multiplying the powers we get,
\[ \Rightarrow {\left( {27{x^6}{y^{12}}} \right)^{\dfrac{2}{3}}} = \left( {{3^2}} \right) \times \left( {{x^4}} \right) \times \left( {{y^8}} \right)\] ,
Now simplifying by multiplying the terms we get,
\[ \Rightarrow {\left( {27{x^6}{y^{12}}} \right)^{\dfrac{2}{3}}} = 9{x^4}{y^8}\] ,
So the given expression is equal to \[9{x^4}{y^8}\] .

The simplified form of given expression \[{\left( {27{x^6}{y^{12}}} \right)^{\dfrac{2}{3}}}\] will be equal to \[9{x^4}{y^8}\] .

Note: There are various laws of exponents we should remember and practise in order to solve and understand the exponential concept. The following are some of the exponent laws:
\[{a^0} = 1\] ,
\[{a^m} \times {a^n} = {a^{m + n}}\] ,
\[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\] ,
\[\dfrac{1}{{{a^m}}} = {a^{ - m}}\] ,
\[{a^m} \times {b^m} = {\left( {ab} \right)^m}\] ,
\[\dfrac{{{a^m}}}{{{b^m}}} = {\left( {\dfrac{a}{b}} \right)^m}\] .