Question

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

Answer 1

$5 \: + \: \frac{2 \sqrt{3} }{7} \: + \: 4 \sqrt{3} \: = \: a \: + \: b \sqrt{3}$
$(5) \: + \: ( \frac{30}{7} ) \sqrt{3} \: = \: a \: + \: b \sqrt{3}$
Hence, by comparing factors on both sides,
a = 5, b = 30/7.

Answer 2

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