Question

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

Answer 1

Hint: We will first rationalize the denominator by multiplying and dividing the given expression by $2 + \sqrt 3 $, then we will just use the general identities of ${a^2} - {b^2} = (a - b)(a + b)$ and ${(a + b)^2} = {a^2} + {b^2} + 2ab$.

Complete step-by-step answer:
We are given that we are required to simplify \[\dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}\].
We will multiply and divide the above mentioned expression by $2 + \sqrt 3 $ to obtain the following expression:-
\[ \Rightarrow \dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }} \times \dfrac{{2 + \sqrt 3 }}{{2 + \sqrt 3 }}\]
Now, we can write this expression as follows:-
\[ \Rightarrow \dfrac{{{{\left( {2 + \sqrt 3 } \right)}^2}}}{{\left( {2 - \sqrt 3 } \right)\left( {2 + \sqrt 3 } \right)}}\] ………………….(1)
Now, since we have an identity given by the following expression: ${(a + b)^2} = {a^2} + {b^2} + 2ab$ for any a and b.
Using this in the numerator of the expression mentioned in the equation number 1, we will then obtain the following equation:-
\[ \Rightarrow \dfrac{{{2^2} + {{\left( {\sqrt 3 } \right)}^2} + 2 \times 2 \times \sqrt 3 }}{{\left( {2 - \sqrt 3 } \right)\left( {2 + \sqrt 3 } \right)}}\]
Simplifying the calculations in the above expression, we will then obtain the following expression:-
\[ \Rightarrow \dfrac{{4 + 3 + 4\sqrt 3 }}{{\left( {2 - \sqrt 3 } \right)\left( {2 + \sqrt 3 } \right)}}\]
Simplifying the numerator by adding the rational numbers in the expression in the above line to obtain the following equation:-
\[ \Rightarrow \dfrac{{7 + 4\sqrt 3 }}{{\left( {2 - \sqrt 3 } \right)\left( {2 + \sqrt 3 } \right)}}\] ………………….(2)
Now, since we have an identity given by the following expression: ${a^2} - {b^2} = (a - b)(a + b)$.
Using this in the denominator of the expression mentioned in the equation number 2, we will then obtain the following equation:-
\[ \Rightarrow \dfrac{{7 + 4\sqrt 3 }}{{{2^2} - {{\left( {\sqrt 3 } \right)}^2}}}\]
Simplifying the calculations in the above expression, we will then obtain the following expression:-
\[ \Rightarrow \dfrac{{7 + 4\sqrt 3 }}{{4 - 3}}\]
Simplifying the calculations further in the above expression, we will then obtain the following expression:-
\[ \Rightarrow 7 + 4\sqrt 3 \]

Hence, we have: \[\dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }} = 7 + 4\sqrt 3 \].

Note:
The students must note that when we multiplied and divided the given expression by $2 + \sqrt 3 $, we have basically multiplied the given expression by 1, because we know that multiplying any quantity by 1 does not affect it at all because 1 is the multiplicative identity.
In other words, we have basically done:
Step 1: Multiply the given expression by 1:
\[ \Rightarrow \dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }} \times 1\]
Step 2: We can write it as follows:
\[ \Rightarrow \dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }} \times \dfrac{{2 + \sqrt 3 }}{{2 + \sqrt 3 }}\]
Now we can just go on as we did above.