Question

If 'a' and 'b' are rational numbers, then find the values of 'a' and 'b' given that $\frac{2+5\sqrt{3}}{2-\sqrt{3}}=a+b\sqrt{3}$

• A. a = 19 , b = 12
• B. a = 12 , b = 19
• C. a = 20 , b = 13
• D. a = 9 , b = 2

Answer 1

The correct option is A

a = 19 , b = 12

$\frac{2+5\sqrt{3}}{2-\sqrt{3}}=a+b\sqrt{3}$

$\frac{2+5\sqrt{3}}{2-\sqrt{3}}\times \frac{2+\sqrt{3}}{2+\sqrt{3}}=a+b\sqrt{3}$

$\frac{2(2+\sqrt{3})+5\sqrt{3}(2+\sqrt{3})}{(2{)}^2-(\sqrt{3}{)}^2}=a+b\sqrt{3}$

$\frac{4+2\sqrt{3}+10\sqrt{3}+15}{4-3}=a+b\sqrt{3}$

$19+12\sqrt{3}=a+b\sqrt{3}$

Comparing we get, $a=19,b=12$