Question

Find the multiplicative inverse of: \[\dfrac{2-\sqrt{3}}{2+\sqrt{3}}\]

Answer 1

Hint: Multiplicative inverse of a number is a number by which the multiplication results in \[1\]. First we have to find the inverse number. In this question we have to multiply \[\dfrac{2-\sqrt{3}}{2+\sqrt{3}}\] with its inverse number, which is \[\dfrac{2+\sqrt{3}}{2-\sqrt{3}}\]. If we multiply them, they nullify each other. After multiplication we get the value will be \[1\] . Then we can proceed with the problem.

Complete step-by-step answer:
We know that inverse means the opposite in effect and the multiplicative inverse of a number is a number by which the multiplication results in \[1\].
To find the multiplicative inverse we have to divide 1 by the number. Let us assume that the number as \[2\] then the multiplicative inverse of the number is \[\dfrac{1}{2}\].
Then according to the definition \[2\times \dfrac{1}{2}=1\]
Here in the question we have to find the multiplicative inverse of \[\dfrac{2-\sqrt{3}}{2+\sqrt{3}}\].
We have to multiply \[\dfrac{2-\sqrt{3}}{2+\sqrt{3}}\] with \[\dfrac{2+\sqrt{3}}{2-\sqrt{3}}\] so that numerator multiply with denominator like \[\left( 2-\sqrt{3} \right)\times \dfrac{1}{\left( 2-\sqrt{3} \right)}\] and \[\left( 2+\sqrt{3} \right)\times \dfrac{1}{\left( 2+\sqrt{3} \right)}\] . After multiplication they nullify each other.
Then, we can finally write it as
\[\dfrac{2-\sqrt{3}}{2+\sqrt{3}}\times \dfrac{2+\sqrt{3}}{2-\sqrt{3}}\]
\[=1\]
Now we find the multiplicative inverse of \[\dfrac{2-\sqrt{3}}{2+\sqrt{3}}\] is \[\dfrac{2+\sqrt{3}}{2-\sqrt{3}}\].

Note: Like this question, students get confused about what is called an inverse of a number, so we have to remember if we divide by the number we get the inverse value of that number. There is another concept called additive inverse, which basically means that if we add additive inverse of the number to the number, then we get the result as 0. Unlike the case of multiplicative inverse, where we are supposed to get the result as 1.