Question

if 8 +√5/8-√5. +. 8-√5/8+√5=a + b √5 find a and bgh

Answer 1

Hello,

The first step is to rationalize the terms in LHS.
this can be done by the method mentioned below :-

$\frac{8 + \sqrt{5} }{8 - \sqrt{5} } + \frac{8 - \sqrt{5} }{8 + \sqrt{5} } \\ \\ = \frac{ {(8 + \sqrt{5} )}^{2} + {(8 - \sqrt{5} )}^{2} }{(8 + \sqrt{5} )(8 - \sqrt{5}) } \\ \\ = \frac{64 + 5 + 16 \sqrt{5} + 64 + 5 - 16 \sqrt{5} }{64 - 5} \\ \\ = \frac{69 + 69}{59} \\ \\ = \frac{138}{59}$
So,
This means that

$\frac{138}{59} = a + b \sqrt{5} \\ \\ a = \frac{138}{59} \\ \\ b = 0$

Hope this will be helping you.....

Answer 2

Answer:

a=$\frac{138}{59}$ and b=0

Step-by-step explanation:

The given equation is:

$\frac{8+\sqrt{5}}{8-\sqrt{5}}+\frac{8-\sqrt{5}}{8+\sqrt{5}}=a+b\sqrt{5}$

On solving the LHS of the above equation, we get

$\frac{8+\sqrt{5}}{8-\sqrt{5}}+\frac{8-\sqrt{5}}{8+\sqrt{5}}$

=$\frac{(8+\sqrt{5})^{2}+(8-\sqrt{5})^{2}}{64-5}$

=$\frac{64+5+16\sqrt{5}+64+5-16\sqrt{5}}{59}$

=$\frac{138}{59}$

Now, comparing with RHS, we get

a=$\frac{138}{59}$ and b=0