Question

if
$\frac{4 + 3 \sqrt{3} }{ \sqrt{7 + 4 \sqrt{3} } } = a + \sqrt{b}$
then b=?​

Answer 1

Given :    ( 4 + 3√3 ) / ( √(7 + 4√3))   = a + √b

To Find : b

Solution:

( 4 + 3√3 ) / ( √(7 + 4√3))   = a + √b

7 + 4√3  = 4 + 3 + 4√3   = 2²  + (√3)²  + 2 .2 (√3)  = (2 + √3)²

√(7 + 4√3) = 2 + √3

( 4 + 3√3 )  / ( 2 + √3)  =  a + √b

Rationalizing LHS

=  (4 + 3√3)  * ( 2 - √3)   /  ( 2 + √3)   ( 2  - √3)  

=  ( 8  - 9  - 4√3 + 6√3) / ( 4 - 3)

= ( - 1 + 2√3)/1

=  - 1 + 2√3

= - 1 + √12

comparing with  a + √b

=> a = - 1  ,  b  = 12

Hence b = 12

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