Question

if 3+√5/2√5+3 = a+b√5 find the value of rational number a and b​

Answer 1

Solution:-

Given

$\rm \implies \dfrac{3 + \sqrt{5} }{3 + 2 \sqrt{5} } = a + b\sqrt{5}$

To find tha value of a and b

Now take

$\rm \implies \: \dfrac{3 + \sqrt{5} }{3 + 2 \sqrt{5} }$

Using rationalization methods

$\rm \implies \: \dfrac{3 + \sqrt{5} }{3 + 2 \sqrt{5} } \times \dfrac{3 - 2 \sqrt{5} }{3 - 2 \sqrt{5} }$

$\rm \implies \: \dfrac{(3 + \sqrt{5})(3 - 2 \sqrt{5} )}{(3 + 2 \sqrt{5})(3 - 2 \sqrt{5} )}$

$\rm \implies \: \dfrac{3 \times 3 - 3 \times 2 \sqrt{5} + 3 \times \sqrt{5} - 2 \sqrt{5} \times \sqrt{5} }{(3) {}^{2} -(2 \sqrt{5})^{2} }$

$\rm \implies \: \dfrac{9 - 6 \sqrt{5} + 3 \sqrt{5} - 10}{9 - 20}$

$\rm \implies \dfrac{ - 1 - 3 \sqrt{5} }{ - 11}$

$\rm \implies \: \dfrac{ - (1 + 3 \sqrt{5}) }{ - 11}$

$\rm \implies \: \dfrac{1}{11} + \dfrac{3}{11} \sqrt{5}$

So value of a

$\rm \implies \: a = \dfrac{1}{11} \: and \: b = \dfrac{3}{11}$