Question

Simplify the given expression: $\dfrac{2}{{\sqrt 5 + \sqrt 3 }} + \dfrac{1}{{\sqrt 3 + \sqrt 2 }} - \dfrac{3}{{\sqrt 5 + \sqrt 2 }}$

Answer 1

Hint: According to given in the question we have to simplify the expression $\dfrac{2}{{\sqrt 5 + \sqrt 3 }} + \dfrac{1}{{\sqrt 3 + \sqrt 2 }} - \dfrac{3}{{\sqrt 5 + \sqrt 2 }}$ So, to simplify the given expression we have to use the rationalization method but first of all let’s understand the rationalization method as explained below:
Rationalization method: Generally the meaning of rationalization is to multiply a rational function by a clever form of one in order to eliminate radical symbols or numbers in the denominator. Rationalization is also a technique used to evaluate limits in order to avoid having a zero or any under root term/number in the denominator when we substitute.
So first of all we will rationalize each part of the given expression then substitute all the terms after rationalization in the expression to add and subtract for the simplification.

Formula used:
$({a^2} - {b^2}) = (a + b) \times (a - b).................(1)$

Complete step by step answer:
Step 1: First of all we have to break the given expressions to rationalize the each term/function of the given expression.
$\dfrac{2}{{\sqrt 5 + \sqrt 3 }} + \dfrac{1}{{\sqrt 3 + \sqrt 2 }} - \dfrac{3}{{\sqrt 5 + \sqrt 2 }}$…………………………..(2)
Step 2: So, here we will rationalize the term $\dfrac{2}{{\sqrt 5 + \sqrt 3 }}$ of the given expression by multiplying $\sqrt 5 - \sqrt 3 $in the numerator and denominator.
$
   = \dfrac{2}{{\sqrt 5 + \sqrt 3 }} \times \dfrac{{\sqrt 5 - \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }} \\
   = \dfrac{{2(\sqrt 5 - \sqrt 3 )}}{{(\sqrt 5 + \sqrt 3 ) \times (\sqrt 5 - \sqrt 3 )}} \\
 $
Now, to simplify the denominator of the obtained expression just above we have to use the formula (1) as mentioned in the solution hint .
$
   = \dfrac{{2(\sqrt 5 - \sqrt 3 )}}{{{{(\sqrt 5 )}^2} - {{(\sqrt 3 )}^2}}} \\
   = \dfrac{{2(\sqrt 5 - \sqrt 3 )}}{{5 - 3}} \\
   = \dfrac{{2(\sqrt 5 - \sqrt 3 )}}{2} \\
   = (\sqrt 5 - \sqrt 3 )................(3) \\
 $
Step 3: Now, we will rationalize the term $\dfrac{1}{{\sqrt 3 + \sqrt 2 }}$ of the given expression by multiplying $\sqrt 3 - \sqrt 2 $in the numerator and denominator.
$ = \dfrac{1}{{\sqrt 3 + \sqrt 2 }} \times \dfrac{{\sqrt 3 - \sqrt 2 }}{{\sqrt 3 - \sqrt 2 }}$
$ = \dfrac{{\sqrt 3 - \sqrt 2 }}{{\sqrt 3 + \sqrt 2 \times \sqrt 3 - \sqrt 2 }}$
Now, to simplify the denominator of the obtained expression just above we have to use the formula (1) as mentioned in the solution hint .
\[
   = \dfrac{{(\sqrt 3 - \sqrt 2 )}}{{{{(\sqrt 3 )}^2} - {{(\sqrt 2 )}^2}}} \\
   = \dfrac{{(\sqrt 3 - \sqrt 2 )}}{{3 - 2}} \\
   = \dfrac{{(\sqrt 3 - \sqrt 2 )}}{1} \\
   = (\sqrt 3 - \sqrt 2 )......................(4) \\
 \]
Step 4: Now, we will rationalize the term $\dfrac{3}{{\sqrt 5 + \sqrt 2 }}$ of the given expression by multiplying $\sqrt 5 - \sqrt 2 $in the numerator and denominator.
$
   = \dfrac{3}{{\sqrt 5 + \sqrt 2 }} \times \dfrac{{\sqrt 5 - \sqrt 2 }}{{\sqrt 5 - \sqrt 2 }} \\
   = \dfrac{{3(\sqrt 5 - \sqrt 2 )}}{{(\sqrt 5 + \sqrt 2 ) \times (\sqrt 5 - \sqrt 2 )}} \\
 $
Now, to simplify the denominator of the obtained expression just above we have to use the formula (1) as mentioned in the solution hint .
\[
   = \dfrac{{3(\sqrt 5 - \sqrt 2 )}}{{{{(\sqrt 5 )}^2} - {{(\sqrt 2 )}^2}}} \\
   = \dfrac{{3(\sqrt 5 - \sqrt 2 )}}{{5 - 2}} \\
   = \dfrac{{3(\sqrt 5 - \sqrt 2 )}}{3} \\
   = (\sqrt 5 - \sqrt 2 )......................(5) \\
 \]
Step 5: On substituting all the rationalized values (3), (4), and (5) in the expression (2)
$
   = (\sqrt 5 - \sqrt 3 ) + (\sqrt 3 - \sqrt 2 ) - (\sqrt 5 - \sqrt 2 ) \\
   = \sqrt 5 - \sqrt 3 + \sqrt 3 - \sqrt 2 - \sqrt 5 + \sqrt 2 \\
 $
Now, on simplifying the expression obtained just above,
$ \Rightarrow \sqrt 5 - \sqrt 3 + \sqrt 3 - \sqrt 2 - \sqrt 5 + \sqrt 2 = 0$
Hence,
$ \Rightarrow \dfrac{2}{{\sqrt 5 + \sqrt 3 }} + \dfrac{1}{{\sqrt 3 + \sqrt 2 }} - \dfrac{3}{{\sqrt 5 + \sqrt 2 }} = 0$

Hence, with the help of the rationalization method and with the help of formula (1) we have simplified the given expression $\dfrac{2}{{\sqrt 5 + \sqrt 3 }} + \dfrac{1}{{\sqrt 3 + \sqrt 2 }} - \dfrac{3}{{\sqrt 5 + \sqrt 2 }} = 0$

Note:
In the rationalization method we have to multiply the inverse of the function/term given in the denominator of the rational number, with the numerator and denominator.
To make the calculation easy we can find the rationalization of each function/term separately and after finding the rationalization of the function/term we can again substitute in the given expression.