If root 2 + root 3 / 3 root 2 - 2 root 3 = a - b root 6 - Maths - Number Systems

Question

If root 2 + root 3 / 3 root 2 - 2 root 3 = a - b root 6 Find
rational numbers a and b

Ankush Jain · Accepted Answer

$G i v e n, \frac{\sqrt{2} + \sqrt{3}}{3 \sqrt{2} - 2 \sqrt{3}} = a - b \sqrt{6}$

On rationalization, we get

$\frac{\sqrt{2} + \sqrt{3}}{3 \sqrt{2} - 2 \sqrt{3}} \times \frac{3 \sqrt{2} + 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}} = a - b \sqrt{6} \Rightarrow \frac{(\sqrt{2} + \sqrt{3}) (3 \sqrt{2} + 2 \sqrt{3})}{18 - 12} = a - b \sqrt{6} \Rightarrow \frac{6 + 3 \sqrt{6} + 2 \sqrt{6} + 6}{6} = a - b \sqrt{6} \Rightarrow \frac{12 + 5 \sqrt{6}}{6} = a - b \sqrt{6} \Rightarrow 2 - (- \frac{5 \sqrt{6}}{6}) = a - b \sqrt{6}$

Comparing the constant terms and coefficient of √6, we get

a = 2 and b = $\frac{- 5}{6}$

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If root 2 + root 3 / 3 root 2 - 2 root 3 = a - b root 6. Find rational numbers a and b.