Question

How do you evaluate the following expression \[\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}\] using the properties of indices?

Answer 1

Hint: This type of problem is based on the concept of properties of indices. First, we have to consider the denominator of the given expression. Use the property of indices that is \[{{a}^{n}}{{a}^{m}}={{a}^{n+m}}\], simplify the denominator. Here a=\[\dfrac{2}{3}\]. Then, substitute the simplified denominator in the expression. And use the property of indices \[\dfrac{{{a}^{n}}}{{{a}^{m}}}={{a}^{n-m}}\] to the expression. Do necessary calculation and simplify the expression. We know that \[{{\left( \dfrac{a}{b} \right)}^{n}}=\dfrac{{{a}^{n}}}{{{b}^{n}}}\], we can find the answer.

Complete step by step solution:
According to the question, we are asked to find the value of \[\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}\].
We have been given the expression is \[\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}\]. ------------(1)
Let us first consider the denominator of the expression (1).
We know that \[{{a}^{n}}{{a}^{m}}={{a}^{n+m}}\]. Using this property of indices, we can simplify the denominator.
\[\Rightarrow {{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}={{\left( \dfrac{2}{3} \right)}^{-5+0}}\]
On further simplification, we get
\[{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}={{\left( \dfrac{2}{3} \right)}^{-5}}\]
Now, let us substitute the simplified denominator to the expression (1).
\[\Rightarrow \dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}=\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}}\]
We know that \[\dfrac{{{a}^{n}}}{{{a}^{m}}}={{a}^{n-m}}\]. Using this property of indices, we get
\[\Rightarrow \dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}={{\left( \dfrac{2}{3} \right)}^{4-\left( -5 \right)}}\]
We know that – (-5) =5. On substituting in the expression, we get
\[\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}={{\left( \dfrac{2}{3} \right)}^{4+5}}\]
On further simplification, we get
\[\Rightarrow \dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}={{\left( \dfrac{2}{3} \right)}^{9}}\]
We have got a simplified expression.
We know that \[{{\left( \dfrac{a}{b} \right)}^{n}}=\dfrac{{{a}^{n}}}{{{b}^{n}}}\]. Using this property, we get
\[\Rightarrow \dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}=\dfrac{{{2}^{9}}}{{{3}^{9}}}\]
We know that \[{{2}^{9}}=512\] and \[{{3}^{9}}=19683\]. On substituting the values in the expression, we get
\[\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}=\dfrac{512}{19683}\]
Therefore, the value of the 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:
We can also solve this problem by another method.
We know that any term to the power 0 is equal to 1. We get
\[\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}=\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}\times 1}\]
\[\Rightarrow \dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}{{\left( \dfrac{2}{3} \right)}^{0}}}=\dfrac{{{\left( \dfrac{2}{3} \right)}^{4}}}{{{\left( \dfrac{2}{3} \right)}^{-5}}}\]
Then, we can simplify the expression as mentioned in the above method. This method reduces the number of steps of the solution. Avoid calculation mistakes based on sign conventions. We should know the properties of indices to solve this type of question.