Question

What is the reciprocal of \[\dfrac{-2}{3}\] ?

Answer 1

Hint: First of all, we need to define what a reciprocal means, it is the multiplicative inverse of any number, Let's say we have a number \[x\] then the reciprocal of this number is given by \[\dfrac{1}{x}\] or we can write it as \[{{x}^{-1}}\]. Multiplicative inverse or reciprocal means that when we multiply two multiplicative inverses, we get 1 as a result.

Complete step by step solution:
The reciprocal word comes from the Latin reciproc(us) which means returning or alternating. Reciprocal actions are based on altering actions or a mutual exchange of something. For finding out the reciprocal we just divide 1 by a given number. Let us understand the meaning of reciprocal more deeply by example.
Now coming to our question
We have
\[x=\dfrac{-2}{3}\]
So, to find the multiplicative inverse of this number we have to find another number which when multiplied to this \[x\] returns us 1.
Now proceeding in the same way
Let reciprocal of \[x\] is \[x'\]
Since \[x\] and \[x'\] are multiplicative inverse, we can say that
\[xx'=1.........\left( i \right)\]
Putting the value of \[x\] in equation\[\left( i \right)\], we get
\[\begin{align}
  & \Rightarrow \dfrac{-2}{3}x'=1 \\
 & \Rightarrow -2x’=3 \\
 & \Rightarrow x’=-\dfrac{3}{2} \\
\end{align}\]

So, in conclusion, reciprocal of \[\dfrac{-2}{3}\] is \[\dfrac{-3}{2}\].

Note: Instead of defining another variable, we can directly reciprocate the number by dividing 1 by the number. Note that the number we reciprocate comes in the denominator so it should not be equals to 0, otherwise the reciprocal would not be defined. In other words reciprocal of zero expression is not defined.