Answer
414.9k+ views
Hint: To solve the question, the first step that we will do is to rationalize the given expression. This involves multiplying both numerator and denominator by the same irrational number. This results in easy simplification further.
Complete step-by-step solution:
The given expression is:
\[\dfrac{{\sqrt 6 + \sqrt 3 }}{{\sqrt 6 - \sqrt 3 }}\]
Here we will use identity to rationalize the denominator of the given expression.
The identity is;
$(a+b)(a-b)={a^2} - {b^2}$.
Here the denominator is \[\sqrt 6 - \sqrt 3 \] which is of the form $(a-b)$. So, we will multiply both numerator and denominator by \[\sqrt 6 + \sqrt 3 \] to get the form of the above identity in the denominator.
So, on multiplying \[\sqrt 6 + \sqrt 3 \] with numerator and denominator we get,
\[\dfrac{{(\sqrt 6 + \sqrt 3 )(\sqrt 6 + \sqrt 3 )}}{{(\sqrt 6 - \sqrt 3 )(\sqrt 6 + \sqrt 3 )}} = \dfrac{{{{(\sqrt 6 + \sqrt 3 )}^2}}}{{{{(\sqrt 6 )}^2} - {{(\sqrt 3 )}^2}}}\]
Using the identity (a+b)(a-b)=${a^2} - {b^2}$ and ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$, the above expression can be simplified as:
\[\dfrac{{6 + 3 + 2\sqrt {18} }}{{6 - 3}} = \dfrac{{9 + 2\sqrt {18} }}{3} = \dfrac{{9 + 2\sqrt {9 \times 2} }}{3} = \dfrac{{3(3 + 2\sqrt 2 )}}{3} = 3 + 2\sqrt 2 \]
Therefore, the simplified expression is \[3 + 2\sqrt 2 \].
Note: You should know about rationalization. It is the process of eliminating a radical or imaginary number from the denominator of an algebraic function by multiplying the same factor in the numerator and denominator. You should know that $\sqrt a \times \sqrt b = \sqrt {ab}$ where a, b are positive numbers.
Complete step-by-step solution:
The given expression is:
\[\dfrac{{\sqrt 6 + \sqrt 3 }}{{\sqrt 6 - \sqrt 3 }}\]
Here we will use identity to rationalize the denominator of the given expression.
The identity is;
$(a+b)(a-b)={a^2} - {b^2}$.
Here the denominator is \[\sqrt 6 - \sqrt 3 \] which is of the form $(a-b)$. So, we will multiply both numerator and denominator by \[\sqrt 6 + \sqrt 3 \] to get the form of the above identity in the denominator.
So, on multiplying \[\sqrt 6 + \sqrt 3 \] with numerator and denominator we get,
\[\dfrac{{(\sqrt 6 + \sqrt 3 )(\sqrt 6 + \sqrt 3 )}}{{(\sqrt 6 - \sqrt 3 )(\sqrt 6 + \sqrt 3 )}} = \dfrac{{{{(\sqrt 6 + \sqrt 3 )}^2}}}{{{{(\sqrt 6 )}^2} - {{(\sqrt 3 )}^2}}}\]
Using the identity (a+b)(a-b)=${a^2} - {b^2}$ and ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$, the above expression can be simplified as:
\[\dfrac{{6 + 3 + 2\sqrt {18} }}{{6 - 3}} = \dfrac{{9 + 2\sqrt {18} }}{3} = \dfrac{{9 + 2\sqrt {9 \times 2} }}{3} = \dfrac{{3(3 + 2\sqrt 2 )}}{3} = 3 + 2\sqrt 2 \]
Therefore, the simplified expression is \[3 + 2\sqrt 2 \].
Note: You should know about rationalization. It is the process of eliminating a radical or imaginary number from the denominator of an algebraic function by multiplying the same factor in the numerator and denominator. You should know that $\sqrt a \times \sqrt b = \sqrt {ab}$ where a, b are positive numbers.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)