Question

plz answer me as soon as possible. plz give me right solution. ..​

Answer 1

$\Large{\underline{\underline{\mathfrak{\bf{Solution}}}}}$

$\Large{\underline{\mathfrak{\bf{Given}}}}$

$\bigstar\sf{\:a\:=\:\dfrac{(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})}} \\ \\ \bigstar\sf{\:b\:=\:\dfrac{(\sqrt{3}+\sqrt{2})}{(\sqrt{3}-\sqrt{2})}}$

$\Large{\underline{\mathfrak{\bf{Find}}}}$

$\mapsto\sf{\:(a^2+b^2-5ab)}$

$\Large{\underline{\mathfrak{\bf{\pink{Explanation}}}}}$

$\mapsto\sf{\green{\:a\:=\:\dfrac{(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})}}}$

( Rationalize of denominator )

$\mapsto\sf{\:a\:=\:\dfrac{(\sqrt{3}-\sqrt{2})(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}} \\ \\ \mapsto\sf{\:a\:=\:\dfrac{(\sqrt{3})^2+(\sqrt{2})^2-2.\sqrt{3}.\sqrt{2}}{(\sqrt{3})^2-(\sqrt{2})^2}} \\ \\ \mapsto\sf{\:a\:=\:\dfrac{(3+2-2\sqrt{6})}{(3-2)}} \\ \\ \mapsto\sf{\:a\:=\:\dfrac{(5-2\sqrt{6})}{1}} \\ \\ \mapsto\sf{\red{\:a\:=\:(5-2\sqrt{6})}}$

Squaring both side,

$\mapsto\sf{\:a^2\:=\:(5-2\sqrt{6})^2} \\ \\ \mapsto\sf{\:a^2\:=\:(25+4.6-2.5.2\sqrt{6})} \\ \\ \mapsto\sf{\:a^2\:=\:(25+24-20\sqrt{6})} \\ \\ \mapsto\sf{\:a^2\:=\:(49-20\sqrt{6})}$

Again,

$\mapsto\sf{\pink{\:b\:=\:\dfrac{(\sqrt{3}+\sqrt{2})}{(\sqrt{3}-\sqrt{2})}}}$

( Rationalize of denominator )

$\mapsto\sf{\:b\:=\:\dfrac{(\sqrt{3}+\sqrt{2})(\sqrt{3}+\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}} \\ \\ \mapsto\sf{\:b\:=\:\dfrac{(\sqrt{3})^2+(\sqrt{2})^2+2.\sqrt{3}.\sqrt{2}}{(\sqrt{3})^2-(\sqrt{2})^2}} \\ \\ \mapsto\sf{\:b\:=\:\dfrac{(3+2+2\sqrt{6})}{(3-2)}} \\ \\ \mapsto\sf{\red{\:b\:=\:(5+2\sqrt{6})}}$

Squaring both side,

$\mapsto\sf{\:b^2\:=\:(5+2\sqrt{6})^2} \\ \\ \mapsto\sf{\:b^2\:=\:(25+4.6+2.5.2\sqrt{6})} \\ \\ \mapsto\sf{\:b^2\:=\:(25+24+20\sqrt{6})} \\ \\ \mapsto\sf{\:b^2\:=\:(49+20\sqrt{6})}$

Now, Calculate here,

$\mapsto\sf{\red{\:(a^2+b^2-5ab)}}$

Keep all Values ,

$\mapsto\sf{\:(49-20\sqrt{6})+(49+20\sqrt{6})-5.(5-2\sqrt{6}).(5+2\sqrt{6})} \\ \\ \mapsto\sf{\:49-20\sqrt{6}+49+20\sqrt{6}-5.(25-24)} \\ \\ \mapsto\sf{\:98-5} \\ \\ \mapsto\bold{\sf{\orange{\:93\:\:\:\:Ans}}}$

Answer 2

Answer:

93

Step-by-step explanation:

Given,

     a = ( √3 - √2 ) / ( √3 + √2 )

     b = ( √3 + √2 ) / ( √3 - √2 )

For the simplification of a :

     Multiplying as well as dividing by ( √3 - √2 ) on RHS of a :

⇒ a = ( √3 - √2 )( √3 - √2 ) / ( √3 + √2 )( √3 - √2 )

⇒ a = ( √3 - √2 )^2 / { ( √3 )^2 - ( √2 )^2 }       { Using ( a + b )( a - b ) = a^2 - b^2 }

⇒ a = ( √3 - √2 )^2 / ( 3 - 2 )

⇒ a = ( √3 - √2 )^2 / 1 = ( √3 - √2 )^2        ...( 1 )

     In the same multiplying as well as dividing RHS of b by √3 + √2

⇒ b = ( √3 + √2 ) / ( √3 - √2 )

⇒ b = ( √3 + √2 )( √3 + √2 ) / ( √3 - √2 )( √3 + √2 )

⇒ b = ( √3 + √2 )^2 / { ( √3 )^2 - ( √2 )^2 }       { Using ( a + b )( a -b ) = a^2 - b^2 }

⇒ b = ( √3 + √2 )^2 / ( 3 - 2 )

⇒ b = ( √3 + √2 )^2 / 1 = ( √3 + √2 )^2       ...( 2 )

 Then, for ab :

From the given, we can say a = 1 / b   or   b = 1 / a  

  Therefore, ab = 1        ..( 3 )

Thus,

⇒ a^2 + b^2 - 5ab

⇒ { ( √3 + √2 )^2 }^2 + { ( √3 - √2 )^2 }^2 - 5( 1 )

⇒ ( 3 + 2 + 2√6 )^2 + ( 3 + 2 - 2√6 )^2 - 5

⇒ ( 5 + 2√6 )^2 + ( 5 - 6√2 )^2 - 5

⇒ ( 25 + 4( 6 )  + 20√6 +  25 + 4( 6 ) - 20√6 - 5

⇒ 50 + 24 + 24 - 5

⇒ 98 - 5

⇒ 93