Question

Rationalize the denominator
$1 \div \sqrt{7 + \sqrt{} 6 + \sqrt{ } 13}$
​

Answer 1

Correct Statement is

Rationalize the denominator

$\sf \: \dfrac{1}{ \sqrt{7} + \sqrt{6} + \sqrt{13} }$

Answer

Basic Concept Used :-

Rationalisation : 

• It is the process by which the square root term or any complicated term in the denominator of a fraction is reduced by multiplying the numerator and denominator of the fraction by the same term as denominator but opposite in sign.

Let's solve the problem now!!

Consider,

$\rm :\longmapsto\: \sf \: \dfrac{1}{ \sqrt{7} + \sqrt{6} + \sqrt{13} }$

$\sf \: \: \: = \: \: \sf \: \dfrac{1} {\bigg(\sqrt{7} + \sqrt{6} \bigg) + \sqrt{13}}$

On rationalizing the denominator, we get

$\sf \: = \: \dfrac{1}{\bigg(\sqrt{7} + \sqrt{6} \bigg) + \sqrt{13} } \times \dfrac{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }$

$\sf \: \: \: = \: \: \dfrac{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }{\bigg(\sqrt{7} + \sqrt{6} \bigg)^{2} - (\sqrt{13})^{2} }$

$\: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \sf \because \: (x + y)(x - y) = {x}^{2} - {y}^{2} }$

$\sf \: \: \: = \: \: \dfrac{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }{7 + 6 + 2 \times \sqrt{7} \times \sqrt{6} - 13}$

$\: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \sf \because \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy}$

$\sf \: \: \: = \: \: \dfrac{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }{ \cancel{13} + 2 \sqrt{42} - \cancel{13}}$

$\sf \: \: \: = \: \: \dfrac{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }{2 \sqrt{42} } \times \dfrac{ \sqrt{42} }{ \sqrt{42} }$

$\sf \: \: \: = \: \: \dfrac{ \sqrt{7 \times 42} + \sqrt{6 \times 42} - \sqrt{13 \times 42} }{2 \times 42}$

$\sf \: \: \: = \: \: \dfrac{7 \sqrt{6} + 6 \sqrt{7}- \sqrt{546} }{84}$

$\sf \: \: \: = \: \: \dfrac{7 \sqrt{6} }{84} + \dfrac{6 \sqrt{7} }{84} - \dfrac{ \sqrt{546} }{84}$

$\sf \: \: \: = \: \: \dfrac{\sqrt{6} }{12} + \dfrac{\sqrt{7} }{14} - \dfrac{ \sqrt{546} }{84}$

Answer 2

Answer:

Answer

Basic Concept Used :-

Rationalisation :

It is the process by which the square root term or any complicated term in the denominator of a fraction is reduced by multiplying the numerator and denominator of the fraction by the same term as denominator but opposite in sign.

Let's solve the problem now!!

Consider,

\rm :\longmapsto\: \sf \: \dfrac{1}{ \sqrt{7} + \sqrt{6} + \sqrt{13} }:⟼

7

+

6

+

13

1

\sf \: \: \: = \: \: \sf \: \dfrac{1} {\bigg(\sqrt{7} + \sqrt{6} \bigg) + \sqrt{13}}=

(

7

+

6

)+

13

1

On rationalizing the denominator, we get

\sf \: = \: \dfrac{1}{\bigg(\sqrt{7} + \sqrt{6} \bigg) + \sqrt{13} } \times \dfrac{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }=

(

7

+

6

)+

13

1

×

(

7

+

6

)−

13

(

7

+

6

)−

13

\sf \: \: \: = \: \: \dfrac{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }{\bigg(\sqrt{7} + \sqrt{6} \bigg)^{2} - (\sqrt{13})^{2} }=

(

7

+

6

)

2

−(

13

)

2

(

7

+

6

)−

13

\: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \sf \because \: (x + y)(x - y) = {x}^{2} - {y}^{2} }

∵(x+y)(x−y)=x

2

−y

2

\sf \: \: \: = \: \: \dfrac{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }{7 + 6 + 2 \times \sqrt{7} \times \sqrt{6} - 13}=

7+6+2×

7

×

6

−13

(

7

+

6

)−

13

\: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \sf \because \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy}

∵(x+y)

2

=x

2

+y

2

+2xy

\sf \: \: \: = \: \: \dfrac{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }{ \cancel{13} + 2 \sqrt{42} - \cancel{13}}=

13

+2

42

−

13

(

7

+

6

)−

13

\sf \: \: \: = \: \: \dfrac{\bigg(\sqrt{7} + \sqrt{6} \bigg) - \sqrt{13} }{2 \sqrt{42} } \times \dfrac{ \sqrt{42} }{ \sqrt{42} }=

2

42

(

7

+

6

)−

13

×

42

42

\sf \: \: \: = \: \: \dfrac{ \sqrt{7 \times 42} + \sqrt{6 \times 42} - \sqrt{13 \times 42} }{2 \times 42}=

2×42

7×42

+

6×42

−

13×42

\sf \: \: \: = \: \: \dfrac{7 \sqrt{6} + 6 \sqrt{7}- \sqrt{546} }{84}=

84

7

6

+6

7

−

546

\sf \: \: \: = \: \: \dfrac{7 \sqrt{6} }{84} + \dfrac{6 \sqrt{7} }{84} - \dfrac{ \sqrt{546} }{84}=

84

7

6

+

84

6

7

−

84

546

\sf \: \: \: = \: \: \dfrac{\sqrt{6} }{12} + \dfrac{\sqrt{7} }{14} - \dfrac{ \sqrt{546} }{84}=

12

6

+

14

7

−

84

546