Question

find 'a' and 'b' it 5 + 2 √3 by 7 + 4 √3 = a + b √3

Answer 1

Heya !!!

Solution:- 5 + 2✓3 / 7 + 4✓3 = A + B✓3

LHS = 5+2✓3/7+4✓3

Rationalizing the denominator 7 + 4✓3

=> 5 + 2✓3 / 7 + 4✓3 × 7 - 4✓3/ 7 -4✓3

=> ( 5 + 2✓3) ( 7 - 4✓3) / ( 7 + 4✓3)(7-4✓3)

=> 5( 7 - 4✓3) + 2✓3( 7 - 4✓3) / (7)² - (4✓3)²

=> 35 - 20✓3 + 14✓3 - 24 / 49 - 48

=> 35 - 24 - 20✓3 + 14✓3/1

=> 11 - 20✓3 +14✓3

=> 11-6✓3

Now,

LHS = RHS

11 - 6✓3 = A + B✓3

Clearly,

A = 11 and B = -6

HOPE IT WILL HELP YOU...... :-)

Answer 2

5+2√3/7+4√3

= 5+2√3/7+4√3×7-4√3/7-4√3

= (5+2√3)(7-4√3)/(7+4√3)(7-4√3)

= 35-20√3+14√3-24/(7)^2-(4√3)^2

= 11-6√3/49-48

= 11-6√3 = a+b√3


a = 11 and b = -6. (On comparing the coefficient)