Question

find a & b such that
$\binom{5 + \sqrt{3} }{7 + 2 \sqrt{3} } = a - b \sqrt{3}$

Answer 1

Heya !!!


5 + ✓3 / 7 + 2✓3 = A - B✓3




LHS = 5+✓3/7+2✓3


Rationalizing the denominator 7+2✓3




=> 5+✓3 / 7 + 2✓3 × ( 7-2✓3)/(7-2✓3)



=> (5+✓3)(7-2✓3)/(7+2✓3)(7-2✓3)



=> 5(7-2✓3)+✓3(7-2✓3)/ ( 7)² - (2✓3)²



=> 35 - 10✓3 + 7✓3 - 6 / 49 - 12



=> 29 - 3✓3/37



Now,



LHS = RHS



29-3✓3/37 = A - B✓3



Clearly,



A = 29/37 and B = 3✓3/47.



★ HOPE IT WILL HELP YOU ★