Question

if 3- root 5 divided by 3+ 2×root 5= a root 5- b, find a and b. (where a and b are rational numbers)​

Answer 1

Answer:

$a = > \frac{9}{11} \: and \: b = > \frac{19}{11}$

Step-by-step explanation:

$Since \: the \: denominator = \\ 3 + 2 \sqrt{5} \\ it's \: rationalizing \: factor = \\ 3 - 2 \sqrt{5} \\ Therefore, \\ \frac{3 - \sqrt{5} }{3 + 2 \sqrt{5} } = \frac{3 - \sqrt{5} }{3 + 2 \sqrt{5} } \times \frac{3 - 2 \sqrt{5} }{3 - 2 \sqrt{5} } \\ = \frac{9 - 6 \sqrt{5} - 3 \sqrt{5} + 10 }{(3) {}^{2} - (2 \sqrt{5} ) {}^{2} } \\ = \frac{19 - 9 \sqrt{5} }{9 - 20} = \frac{19 - 9 \sqrt{5} }{ - 11} \\ = \frac{ -19 + 9 \sqrt{5} }{11} \\ = \frac{9 \sqrt{5} - 19 }{11} = > \frac{9 \sqrt{5} }{11} - \frac{19}{11} \\ Given: \\ \frac{3 - \sqrt{5} }{3 + 2 \sqrt{5} } = a \sqrt{5} - b \\ = > \frac{9 \sqrt{5} }{11} - \frac{19}{11} \\ = > a = \frac{9}{11} \: \: and \: \: b = \frac{19}{11}$