Question

Simplify the following expression
$\dfrac{5+\sqrt{3}}{2-\sqrt{3}}+\dfrac{2-\sqrt{3}}{2+\sqrt{3}}$

Answer 1

Hint: We will rationalize the denominators of both the terms and simplify the equation. To rationalize, we will multiply and divide each fraction by the conjugate of the denominator of that fraction. We will use algebraic identities to obtain a simplified expression of the given expression.

Complete step by step answer:
Let us denote the given expression as $E=\dfrac{5+\sqrt{3}}{2-\sqrt{3}}+\dfrac{2-\sqrt{3}}{2+\sqrt{3}}$. Now, will look at the first term of the given equation. To rationalize the first term, we will multiply and divide the fraction by the conjugate of $2-\sqrt{3}$, which is $2+\sqrt{3}$. So, we get
$\dfrac{5+\sqrt{3}}{2-\sqrt{3}}=\dfrac{5+\sqrt{3}}{2-\sqrt{3}}\times \dfrac{2+\sqrt{3}}{2+\sqrt{3}}$.
We know that $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$. Using this algebraic identity, the denominator will become $\left( 2-\sqrt{3} \right)\left( 2+\sqrt{3} \right)={{2}^{2}}-{{\left( \sqrt{3} \right)}^{2}}=4-3=1$. Now, the first term has only the numerator, which is $\left( 5+\sqrt{3} \right)\left( 2+\sqrt{3} \right)=10+5\sqrt{3}+2\sqrt{3}+3=13+7\sqrt{3}$.
Now, we will look at the second term of the given expression. The second term of $E$ is $\dfrac{2-\sqrt{3}}{2+\sqrt{3}}$. To rationalize this fraction, we will multiply and divide it by the conjugate of the denominator $2+\sqrt{3}$, which is $2-\sqrt{3}$. The denominator of the second term will become the same as the denominator of the first term, since the radicals involved in the multiplication are the same. So, the denominator is $1$. Now, the numerator will be $\left( 2-\sqrt{3} \right)\left( 2-\sqrt{3} \right)={{\left( 2-\sqrt{3} \right)}^{2}}=4-4\sqrt{3}+3=7-4\sqrt{3}$.
We have the simplified forms of the first and the second term of the given expression $E$. Substituting these simplified forms in $E$, we get
$\begin{align}
  & E=13+7\sqrt{3}+7-4\sqrt{3} \\
 & =20+3\sqrt{3}
\end{align}$

The simplified form of the given expression is $20+3\sqrt{3}$.

Note: It is important to be familiar with the concept of rationalization. It is very useful in simplifying complicated expressions. Algebraic identities are also a big help and save time while solving such types of questions. To avoid confusion, it is convenient to deal with individual terms of the given expression.