Question

2 + 3 root 3/4 - 5 root 3 = a+b root 3​

Answer 1

$\frac{2 + 3 \sqrt{3} }{4 - 5 \sqrt{3} } = a + b \sqrt{3} \\ \\ \frac{2 + 3 \sqrt{3} }{4 - 5 \sqrt{3} } \times \frac{4 + 5 \sqrt{3} }{4 + 5 \sqrt{3} } = a + b \sqrt{3} \\ \\ \frac{8 + 10 \sqrt{3} + 12 \sqrt{3} + 45}{ {4}^{2} - ( {5 \sqrt{3} })^{2} } = a + b \sqrt{3} \\ \\ \frac{53 + 22 \sqrt{3} }{ - 59} = a + b \sqrt{3} \\ \\ - \frac{53}{59} - \frac{22}{59} \sqrt{3} = a + b \sqrt{3} \\ \\ comparing \: both \: sides \\ a = - \frac{53}{59} \\ \\ b = - \frac{22}{59}$