Question

Rationalize the equation \[\dfrac{{\sqrt 3 - 1}}{{\sqrt 3 + 1}}\]

Answer 1

Hint: To solve the above question you need to know about the concept of rationalization of denominator. The elimination of radicals in the denominator of an algebraic fraction is known as root rationalization. In the above question, we will multiply and divide the denominator to get the desired answer.

Formula used:
To solve the above question, keep in mind the formula of \[\left( {{a^2} - {b^2}} \right)\].
Here, \[\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)\].

Complete step-by-step answer:
We have to rationalize: \[\dfrac{{\sqrt 3 - 1}}{{\sqrt 3 + 1}}\]
Now, we will rationalize the equation by multiply and dividing the denominator by \[\sqrt 3 - 1\].
On solving, we get: \[\dfrac{{\sqrt 3 - 1}}{{\sqrt 3 + 1}} \times \dfrac{{\sqrt 3 - 1}}{{\sqrt 3 - 1}}\]
Now use the formula \[\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)\] in the denominator.
We get:
\[
  \dfrac{{{{\left( {\sqrt 3 - 1} \right)}^2}}}{{{{\left( {\sqrt 3 } \right)}^2} - {{\left( 1 \right)}^2}}} \\
   \Rightarrow \dfrac{{3 + 1 - 2\sqrt 3 }}{{3 - 1}} \\
   \Rightarrow \dfrac{{4 - 2\sqrt 3 }}{2} \\
 \]
This can be written as:
\[
  \dfrac{{2\left( {2 - \sqrt 3 } \right)}}{2} \\
   \Rightarrow 2 - \sqrt 3 \\
 \].
So, the final answer is \[2 - \sqrt 3 \].

Note: While solving questions similar to the one given above, remember rationalization is a method that is used. It aids in the reduction of mathematical expressions to their simplest form. Making something more powerful is what rationalization is all about. Its mathematical adaptation entails reducing the equation to a more effective and simpler form. Also, in the denominator of an equation, square roots are not allowed. To get rid of the square root, we need to multiply. This is achieved by multiplying the bottom denominator by the top and bottom denominators of the equation. This would remove the square root, resulting in normal numbers in your equation. After you've done that, the equation will be solved, and you'll be able to simplify it to get the final response.