How do you simplify \[\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}\]?

Hint: In order to find the solution of the given question that is to simplify \[\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}\] apply the following formulas of exponential rule that are \[\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}\] and \[{{\left( \dfrac{a}{b} \right)}^{c}}=\dfrac{{{a}^{c}}}{{{b}^{c}}}\]. Also apply the result \[{{a}^{0}}=1,a\ne 0\] to simplify the given expression.

Complete step by step solution:
According to the question, given expression in the question is as follows:
\[\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}\]
To simplify the above expression further, apply the result \[{{a}^{0}}=1,a\ne 0\]which means any non-zero term to the power zero is always equal to one, we will have:
\[\Rightarrow \dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}\cdot 1}\]
\[\Rightarrow \dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}}\]
Now apply the exponential rule in the above expression that is \[\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}\] this means that two terms with same base and different power can be written as that base to the power subtraction of both the given powers if the terms are to divided, we will have:
\[\Rightarrow {{\left( \dfrac{2}{3} \right)}^{4-\left( -5 \right)}}\]
Simplify the above expression by opening the bracket in the power of the fraction with the help of subtraction, we will get:
\[\Rightarrow {{\left( \dfrac{2}{3} \right)}^{9}}\]
After this apply the exponential rule that is \[{{\left( \dfrac{a}{b} \right)}^{c}}=\dfrac{{{a}^{c}}}{{{b}^{c}}}\] in the above expression, we will have:
\[\Rightarrow \dfrac{{{2}^{9}}}{{{3}^{9}}}\]
We know that \[{{2}^{9}}=512\], applying this result in the above expression, we will get:
\[\Rightarrow \dfrac{512}{{{3}^{9}}}\]
We know that \[{{3}^{9}}=19683\], applying this result in the above expression, we will get:
\[\Rightarrow \dfrac{512}{19683}\]

Therefore, simplified form of the given expression \[\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}\] is \[\dfrac{512}{19683}\].

Note: Students make mistakes by applying the wrong exponential rule. They end up applying the exponential rule as \[\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a+b}}\] instead of \[\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}\] . This is completely wrong and leads to the incorrect answer. It’s important to remember the correct exponential rule is \[\dfrac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}\].