How do you simplify \[\dfrac{{{\left( {{x}^{6}} \right)}^{3}}}{{{\left( {{x}^{3}} \right)}^{4}}}\]?

Hint: This type of problem can be solved with the properties of powers. First, we have to consider the numerator of the given function. We can express the given function as \[{{x}^{6}}={{x}^{3\times 2}}\]. Using the rule \[{{a}^{n\times m}}={{\left( {{a}^{n}} \right)}^{m}}\], we can convert the numerator as \[{{\left( {{x}^{3}} \right)}^{2\times 3}}\]. Do necessary calculations and simplify the numerator. Substitute this in the function. Using the division rule of powers, that is \[\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\], we can simplify the given function.

Complete step by step solution:
According to the question, we are asked to simplify the function \[\dfrac{{{\left( {{x}^{6}} \right)}^{3}}}{{{\left( {{x}^{3}} \right)}^{4}}}\].
We have been given the function is \[\dfrac{{{\left( {{x}^{6}} \right)}^{3}}}{{{\left( {{x}^{3}} \right)}^{4}}}\]. --------(1)
We first have to consider the numerator of the function.
We know that \[{{x}^{6}}={{x}^{3\times 2}}\]. Substituting in the numerator, we get
\[{{\left( {{x}^{6}} \right)}^{3}}={{\left( {{x}^{3\times 2}} \right)}^{3}}\]
We know that \[{{a}^{n\times m}}={{\left( {{a}^{n}} \right)}^{m}}\]. Let us take 2 from the power out of the bracket.
\[\Rightarrow {{\left( {{x}^{6}} \right)}^{3}}={{\left( {{\left( {{x}^{3}} \right)}^{2}} \right)}^{3}}\]
Using the same property of powers \[{{\left( {{a}^{n}} \right)}^{m}}={{a}^{n\times m}}\], we get
\[{{\left( {{x}^{6}} \right)}^{3}}={{\left( {{x}^{3}} \right)}^{2\times 3}}\]
On further simplifications, we get
\[{{\left( {{x}^{6}} \right)}^{3}}={{\left( {{x}^{3}} \right)}^{6}}\]
Now, substitute the simplified numerator to the function (1).
\[\dfrac{{{\left( {{x}^{6}} \right)}^{3}}}{{{\left( {{x}^{3}} \right)}^{4}}}=\dfrac{{{\left( {{x}^{3}} \right)}^{6}}}{{{\left( {{x}^{3}} \right)}^{4}}}\]
We find that both the numerator and denominator have same terms with different powers.
Let us use the property \[\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\] to solve further.
We get
\[\dfrac{{{\left( {{x}^{6}} \right)}^{3}}}{{{\left( {{x}^{3}} \right)}^{4}}}={{\left( {{x}^{3}} \right)}^{6-4}}\].
We know that 6-4=2. Let us use this in the above function.
\[\Rightarrow \dfrac{{{\left( {{x}^{6}} \right)}^{3}}}{{{\left( {{x}^{3}} \right)}^{4}}}={{\left( {{x}^{3}} \right)}^{2}}\]
We know that \[{{\left( {{a}^{n}} \right)}^{m}}={{a}^{n\times m}}\]. Using this property in the function, we get
\[\dfrac{{{\left( {{x}^{6}} \right)}^{3}}}{{{\left( {{x}^{3}} \right)}^{4}}}={{x}^{3\times 2}}\]
On further simplification, we get
\[\dfrac{{{\left( {{x}^{6}} \right)}^{3}}}{{{\left( {{x}^{3}} \right)}^{4}}}={{x}^{6}}\]

Therefore, the simplified form of \[\dfrac{{{\left( {{x}^{6}} \right)}^{3}}}{{{\left( {{x}^{3}} \right)}^{4}}}\] is \[{{x}^{6}}\].

Note: Whenever we get such types of questions, we have to separately solve the numerator and denominator and convert them into the same terms with different powers. Avoid calculation mistakes based on sign conventions. We should also know the properties of power to solve this type of question.