Question

Write the reciprocal of each of the following:
\[(1)\]\[{{16}^{-7}}\]
\[(2)\] \[{{\left(\dfrac{2}{3} \right)}^{-4}}\]

Answer 1

Hint: To solve this question we should have the knowledge of the reciprocal of the number. We also require the knowledge of the index of the number and the properties of indices as we need to deal with the same while solving the question. The reciprocal of a number is also known as the multiplicative inverse. We have to apply the concept of reciprocal separately on the subparts of the question to get the required answers.

Complete step-by-step solution:
This question requires knowledge of the reciprocal of a number. As the numbers given in the question are raised to some negative power we should also know the properties of indices. The same concept will be applied on both the subparts and we will reach our answer. Let us now discuss the above concepts one by one.
Properties of indices: We will understand the property required in the question by the following example.
Evaluate \[{{4}^{-2}}\].
The solution to the above question is given as,
      \[{{4}^{-2}}=\frac{1}{{{4}^{2}}}\]
\[\Rightarrow {{4}^{-2}}=\frac{1}{16}\]

Reciprocal of a number: The reciprocal of a number is defined as the number that gives output \[1\] when multiplied to that given number itself. It is also known as the multiplicative inverse of the number. It is one of the most important properties of numbers. In simple words, the output of the reciprocal of the number is a fraction in which the numerator is \[1\] and the denominator is the given number itself. The reciprocal of \[0\] is not defined as \[0\] will be the denominator in that case. The reciprocal of a positive number is always positive and that of the negative number is always negative. The reciprocal of a number greater than \[1\] is smaller than the number itself and that for the number between \[0\] and \[1\] is greater than the number itself.
After revising all the required concepts, let us now move towards the main question.
Let, \[R\]be the reciprocal of the given number.
For first part,
The reciprocal of \[{{16}^{-7}}\] is given by,
      \[R=\frac{1}{{{16}^{-7}}}\]
\[\begin{align}
  & \Rightarrow R=\frac{1}{\frac{1}{{{16}^{7}}}} \\
 & \Rightarrow R={{16}^{7}} \\
 & \Rightarrow R=268435456 \\
\end{align}\]
For second part,
The reciprocal of \[{{\left(\dfrac{2}{3} \right)}^{-4}}\]is given by,
     \[R=\frac{1}{{{\left(\dfrac{2}{3} \right)}^{-4}}}\]
\[\begin{align}
  & \Rightarrow R=\frac{1}{\frac{1}{{{\left(\dfrac{2}{3} \right)}^{4}}}} \\
 & \Rightarrow R={{\left(\dfrac{2}{3} \right)}^{4}} \\
 & \Rightarrow R=\left(\dfrac{16}{81} \right) \\
\end{align}\]
From this we can conclude that the reciprocal of \[{{16}^{-7}}\] is \[268435456\] and that of \[{{\left(\dfrac{2}{3} \right)}^{-4}}\] is \[\frac{16}{81}\].

Note: This question contained two subparts but we need to apply the same concept to both of them. This just requires the knowledge of the reciprocal of a number. The knowledge of properties of indices was also tested by the given question. We should also remember the powers of the number and evaluate them to make the steps to the answer to become easy.