Question

How do you simplify $\dfrac{3}{{6 - \sqrt 5 }}$?

Answer 1

Hint: In order to obtain the simplified form of the above expression , use the method of rationalisation by multiplying and dividing the fraction with $\left( {6 + \sqrt 5 } \right)$ to eliminate the square root part of the denominator. Expand the numerator using distributive property and rewrite the denominator using formula $\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^2}$. Simplify the obtained expression by cancelling out common factors and separate the denominator with the terms of the numerator to obtain the required result.

Formula used:
$\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^2}$

Complete step by step solution:
We are given an expression $\dfrac{3}{{6 - \sqrt 5 }}$
As we can see there is a square root term in the denominator , so to simplify the expression given we have to first eliminate this square root term from the denominator by rationalizing the fraction.
So for this we are going to multiply both numerator and denominator with $\left( {6 + \sqrt 5 } \right)$.
We cannot multiply anything with numerator or denominator only , as this will disbalance the fraction ,so we have to multiply the same number with both numerator and denominator to maintain that balance of the fraction.
Now multiplying both numerator and denominator with the fraction given , we get
\[
   = \dfrac{3}{{6 - \sqrt 5 }} \times \dfrac{{\left( {6 + \sqrt 5 } \right)}}{{\left( {6 + \sqrt 5 } \right)}} \\
   = \dfrac{{3\left( {6 + \sqrt 5 } \right)}}{{\left( {6 - \sqrt 5 } \right)\left( {6 + \sqrt 5 } \right)}} \\
 \]
Now expanding the numerator part with the use of distributive property of multiplication as $A\left( {B + C} \right) = AB + AC$ and rewriting the denominator using the formula $\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^2}$ by considering A as $6$ and B as $\sqrt 5 $.
Our expression now becomes
\[
   = \dfrac{{18 + 3\sqrt 5 }}{{{6^2} - {{\left( {\sqrt 5 } \right)}^2}}} \\
   = \dfrac{{18 + 3\sqrt 5 }}{{36 - 5}} \\
   = \dfrac{{18 + 3\sqrt 5 }}{{31}} \\
 \]
Separating the denominator with the terms of numerator ,we obtain
\[ = \dfrac{{18}}{{31}} + \dfrac{{3\sqrt 5 }}{{31}}\]

Therefore, the simplification of the expression $\dfrac{3}{{6 - \sqrt 5 }}$ is equal to \[\dfrac{{18}}{{31}} + \dfrac{{3\sqrt 5 }}{{31}}\].

Note: 1. Make sure to combine all the like terms in the numerator as well as denominator before separating the denominator with the term of numerator.
2. Don’t forget to cross verify your result in the end.
3. In the rationalisation method, we have to always multiply the inverse of the denominator with both numerator and denominator.