Question

if a and b are rational numbers and

$\frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} } = a + b \sqrt{5}$
find the values of a and b
​

Answer 1

Answer:

a = 61/ 29, b = 24/29.

Step-by-step explanation:

Consider LHS $\frac{4 + 3\sqrt{5} }{4 - 3\sqrt{5} }$

Now to simplify ,rationalize the denominator,

we get

$\frac{4 + 3\sqrt{5} }{4 - 3\sqrt{5} }$  = $\frac{4 + 3\sqrt{5} }{4 - 3\sqrt{5} }$   x  $\frac{4 + 3\sqrt{5} }{4+ 3\sqrt{5} }$

on further solving , we get

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

Using algebraic identity

$(a+b)^{2}  = a^{2}  + b^{2}  + 2ab\\(a+b)(a-b) =  a^{2}  - b^{2}$

we get,

$\frac{16 + 45 + 24\sqrt{5} }{16 - 45}$  

= $\frac{61 + 24\sqrt{5} }{29}$

Compare left side with right side, we get

$\frac{61 + 24\sqrt{5} }{29}$  = a + b√5

So, a = 61/ 29

and b = 24/29.