Question

5 + 2 root 3 / 7 + 4 root 3 equals to a + b root 3

Answer 1

`5 \: + \: (2 √(3) )/(7) \: + \: 4 √(3) \: = \: a \: + \: b √(3)`

`(5) \: + \: ( (30)/(7) ) √(3) \: = \: a \: + \: b √(3)`
Hence, by comparing factors on both sides,
a = 5, b = 30/7.

Answer 2

`(5+2\sqrt3)/(7+4\sqrt3)=(5+2\sqrt3)/(7+4\sqrt3)\cdot(7-4\sqrt3)/(7-4\sqrt3)\\\\\text{use}\ (a+b)(a-b)=a^2-b^2\\\\=((5+2\sqrt3)(7-4\sqrt3))/(7^2-(4\sqrt3)^2)\\\\\text{use distributive property}\\\\=((5)(7)+(5)(-4\sqrt3)+(2\sqrt3)(7)+(2\sqrt3)(-4\sqrt3))/(49-4^2(\sqrt3)^2)\\\\=(35-20\sqrt3+14\sqrt3-8(3))/(49-16(3))=(35-6\sqrt3-24)/(49-48)=(11-6\sqrt3)/(1)\\\\=11-6\sqrt3\\\\11-6\sqrt3=a+b\sqrt3\to\boxed{a=11\ and\ b=-6}`