Question

If a and b are rational numbers and 4+3 roor 5/4-3root5=a+b root 5 find the value of a and b

Answer 1

Given

$\dfrac{4+3\sqrt{5} }{4-3\sqrt{5} } = a+b\sqrt{5}$

LHS

$= \dfrac{4+3\sqrt{5} }{4-3\sqrt{5} }$

$= \dfrac{4+3\sqrt{5} }{4-3\sqrt{5} } \times \dfrac{4+3\sqrt{5} }{4+3\sqrt{5} }$

$= \dfrac{(4+3\sqrt{5})^{2} }{(4)^{2} -(3\sqrt{5})^{2} }$

$= \dfrac{16+24\sqrt{5}+45 }{16-45 }$

$= \dfrac{61+24\sqrt{5} }{-29 }$

$= \dfrac{-61}{29}+\dfrac{-24\sqrt{5} }{29}$

$\boxed{\boxed{\bold{\implies a = \dfrac{-61}{29}}}}}}$

$\boxed{\boxed{\bold{\implies b =\dfrac{-24\sqrt{5} }{29}}}}}}}}}$