Question

How do you simplify $\dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}$ ?

Answer 1

Hint: Here we can simplify the given expression by conjugate. That is, multiply both the numerator and denominator of the fraction by the conjugate of the denominator to clear the square root from the denominator.

Complete step-by-step solution:
Given radical expression is $\dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}$,
The process is the same, regardless, namely.
We flip the sign in the middle. Since it gives an expression with a ‘plus’ in the middle, the conjugate is the same two terms, but with a ‘minus’ in the middle.
Or if it has ‘minus’ we can change the middle ‘plus’
In this equation the denominator $2 - \sqrt 3 $ so conjugate of this term is $2 + \sqrt 3 $,
So multiply both the numerator and denominator of the fraction by the conjugate of the denominator to clear the square root from the denominator.
$ \Rightarrow \dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }} = \dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }} \times \dfrac{{2 + \sqrt 3 }}{{2 + \sqrt 3 }}$ -------(1)
Simplify the denominators which like the factors of difference between to perfect squares so we use the formula $\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)$
Here we have $\left( {2 - \sqrt 3 } \right)\left( {2 + \sqrt 3 } \right) = \left( {{2^2} - {{\left( {\sqrt 3 } \right)}^2}} \right)$ we get,
$ \Rightarrow \left( {4 - 3} \right) = 1$
Substitute the above simplified term into the equation (1) we get,
$ \Rightarrow \dfrac{{2 - \sqrt 3 }}{{2 + \sqrt 3 }} = \dfrac{{\left( {2 + \sqrt 3 } \right)\left( {2 + \sqrt 3 } \right)}}{1}$---------(2)
Now the denominator multiply the term $\left( {2 + \sqrt 3 } \right)\left( {2 + \sqrt 3 } \right) = {\left( {2 + \sqrt 3 } \right)^2}$
So we use the formula ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ we get,
$ \Rightarrow {\left( {2 + \sqrt 3 } \right)^2} = {2^2} + 2\left( 2 \right)\left( {\sqrt 3 } \right) + {\left( {\sqrt 3 } \right)^2}$
Simplifying it we get,
 $ \Rightarrow {\left( {2 + \sqrt 3 } \right)^2} = 4 + 4\sqrt 3 + 3$
Let us add the term and we get
$ \Rightarrow {\left( {2 + \sqrt 3 } \right)^2} = 7 + 4\sqrt 3 $
Substitute the above simplified term into the equation (2) we get,
$ \Rightarrow \dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }} = 7 + 4\sqrt 3 $

$7 + 4\sqrt 3 $ is the required answer.

Note: Perhaps a conjugate’s most useful function is as a tool when simplifying expressions with radicals, or square roots. In this case we are going to find the conjugate for an expression in which only one of the terms has a radical.
In this problem we can easily find the conjugate, but if the radical is in the form of the first of the two terms, and there is a ‘minus’ in front of the first term. We have the first ‘minus’ alone, because we don’t change any but the middle sign.