Question

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

Answer 1

Hint: This problem deals with the basic mathematical concepts, which includes the topic of the reciprocation, or with finding the inverse of any given number. In order to solve this problem, we are doing the given number making into a positive integer and then solving for the reciprocal of a required number. Reciprocal of a given number is the inverse of the number.

Complete step-by-step solution:
The reciprocal of any given number is nothing but the inverse of that particular given number. Reciprocal is also called as multiplicative inverse. Reciprocals are used to make the equations easier to solve.
The general form of a fraction and its reciprocal is \[\dfrac{a}{b}=\dfrac{b}{a}\].
The given numbers are positive numbers with a negative power.
(i)Given that a number which is \[{{16}^{-7}}\], it can be written as \[{{\left( \dfrac{1}{16} \right)}^{7}}\]
By applying the general form of reciprocal, we can get the Reciprocal form of \[{{16}^{-7}}\]as \[{{\left( \dfrac{16}{1} \right)}^{7}}={{\left( 16 \right)}^{7}}\]
Hence, the reciprocal of \[{{16}^{-7}}\]=\[{{\left( 16 \right)}^{7}}\]= 268435456.
(ii)Given that a number which is \[{{\left( \dfrac{2}{3} \right)}^{-4}}\]
By applying the general form of reciprocal, we can get the Reciprocal form of \[{{\left( \dfrac{2}{3} \right)}^{-4}}\]as \[{{\left( \dfrac{3}{2} \right)}^{4}}\]
Hence the reciprocal of \[{{\left( \dfrac{2}{3} \right)}^{-4}} is ={{\left( \dfrac{2}{3} \right)}^{4}}\].

Note: Reciprocal can be found by changing the position of the numerator and the denominator. It can be checked by multiplying the two expressions together and ensuring that you answer is 1. It must be done when the first portion numerator is unique in the relation to zero. And also remember that the reciprocal of a negative number must itself be a negative number.