Question

Find the multiplicative inverse of each of the following: - \[3+\sqrt{8}\].

Answer 1

Hint: Understand the definition of multiplicative inverse of the given number: - \[3+\sqrt{8}\]. Multiply x with \[3+\sqrt{8}\] and equate it with 1. Now, write x as a reciprocal of \[3+\sqrt{8}\] and rationalize the denominator to get the answer.

Complete step-by-step solution
Here, we have to find the multiplicative inverse of the given number: \[3+\sqrt{8}\]. First, let us know the definition of this term.
In mathematics, a multiplicative inverse or reciprocal for a number, denote by \[\dfrac{1}{x}\] or \[{{x}^{-1}}\], is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction \[\dfrac{m}{n}\] is \[\dfrac{n}{m}\], For example: - the multiplicative inverse of 2 is \[\dfrac{1}{2}\].
Now, let us come to the question. We have to find the multiplicative inverse of \[3+\sqrt{8}\]. Let us assume this multiplicative inverse as ‘x’. Therefore, we have,
\[\Rightarrow x\times 3+\sqrt{8}=1\]
\[\Rightarrow x=\dfrac{1}{3+\sqrt{8}}\] - (1)
Now, let us rationalize the right-hand side of the expression.
1. Therefore, we have,
\[\Rightarrow \dfrac{1}{3+\sqrt{8}}=\dfrac{1}{3+\sqrt{8}}\times \dfrac{3-\sqrt{8}}{3-\sqrt{8}}\]
Here, in the above expression we have multiplied \[\dfrac{1}{3+\sqrt{8}}\] with its conjugate \[3-\sqrt{8}\] and also divided it with the same conjugate. So, that the equation get balanced. So, we have,
\[\Rightarrow \dfrac{1}{3+\sqrt{8}}=\dfrac{3-\sqrt{8}}{\left( 3+\sqrt{8} \right)\left( 3-\sqrt{8} \right)}\]
Applying the identity, \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\], in the denominator, we get,
\[\begin{align}
  & \Rightarrow \dfrac{1}{3+\sqrt{8}}=\dfrac{3-\sqrt{8}}{{{3}^{2}}-{{\left( \sqrt{8} \right)}^{2}}} \\
 & \Rightarrow \dfrac{1}{3+\sqrt{8}}=\dfrac{3-\sqrt{8}}{9-8} \\
 & \Rightarrow \dfrac{1}{3+\sqrt{8}}=3-\sqrt{8}=x \\
\end{align}\]
Hence, the multiplicative inverse of \[3+\sqrt{8}\] is \[3-\sqrt{8}\].

Note: One may note that we can check our answer by multiplying the obtained answer with the given number. If we get the value 1 then our answer is correct otherwise not. Remember that while rationalizing the denominator or numerator we multiply the term with its conjugate and also divide it with the same conjugate so that the expression does not change.