Question

(5+√3)/(7-4√3) =a+b√3 , find the value of a and b class 9 maths

Answer 1

Answer :-

$\implies\sf \dfrac{5+\sqrt3}{7-4\sqrt3} = a + b\sqrt3$

Solving LHS :-

$\implies\sf \dfrac{5+\sqrt3}{7-4\sqrt3}$

$\implies\sf \dfrac{5+\sqrt3}{7-4\sqrt3} \times \dfrac{7+4\sqrt3}{7+4\sqrt3}$

$\implies\sf \dfrac{ (5+\sqrt3)(7+4\sqrt3)}{(7-4\sqrt3)(7+4\sqrt3)}$

$\implies\sf \dfrac{5( 7 + 4\sqrt3) + \sqrt3(7 + 4\sqrt3)}{ 7^2 - (4\sqrt3)^2}$

$\implies\sf \dfrac{35 + 20\sqrt3 + 7\sqrt3 + 4 \times 3}{49 - 48}$

$\implies\sf \dfrac{35 + 27\sqrt3 + 12}{1}$

$\implies\sf 47 + 27\sqrt3$

Comparing with RHS :-

$\implies\sf 47 + 27\sqrt3 = a + b\sqrt3$

• a = 47
• b = 27

Value of a = 47 and b = 27

Answer 2

QuestioN :

(5+√3)/(7-4√3) =a+b√3 , find the value of a and b

GiveN :

• (5+√3)/(7-4√3) =a+b√3

To FiNd :

• The value of a and b.

ANswer :

The value of a = 7 , b = - 3.

SolutioN :

√7 + 4√3

= √7 + 2√ ( 4 × 3 )

= √( 4 + 3 ) + 2√( 4 × 3 )

Therefore ,

√7 + 4√3 = √4 + √3

= 2 + √3----( 1 )

LHS = ( 5 + √3 ) / ( √7 - 4√3 )

= ( 5 + √3 ) / ( 2 + √3 )

Rationalize the denominator

= (5+√3 )(2 -√3)/(2 + √3 ) ( 2-√3 )

= [10-5√3+2√3-3]/[( 2 )²- (√3)²]

= ( 7 - 3√3 )/(4 - 3 )

= 7 - 3√3

= RHS

7 - 3√3 = a + b√3

Compare both sides

a = 7 , b = - 3

∴Hence, The value of a = 7 , b = - 3.

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