Question

7 + 3root 5/ 3 + root 5 minus 7 minus 3 root 5/ 3 minus root 5= a+ b root 5

Answer 1

Here is the answer to the question..

Answer 2

Given:

$\bold{\frac{7 + 3\sqrt{5}}{3+ \sqrt{5} } - \frac{7- 3\sqrt{5}}{3- \sqrt{5}} = a+ b\sqrt{5}}$

To find:

value in a and b form.

Solution:

$\Rightarrow \frac{7 + 3\sqrt{5}}{3+ \sqrt{5} } - \frac{7- 3\sqrt{5}}{3- \sqrt{5}} = a+ b\sqrt{5}$

Solve the L.H.S part:

$\Rightarrow \frac{(7 + 3\sqrt{5})(3-\sqrt{5})- (3+ \sqrt{5}) (7- 3\sqrt{5}) }{(3+\sqrt{5}) (3-\sqrt{5})} = a+ b\sqrt{5}\\\\\Rightarrow \frac{(21-7\sqrt{5}+9\sqrt{5}-15)- (21-9\sqrt{5}+7\sqrt{5}-15)}{ ((3)^2-(\sqrt{5})^2)} = a+ b\sqrt{5}\\\\\Rightarrow \frac{(6+2\sqrt{5})- (6-2\sqrt{5})}{(9-5)} = a+ b\sqrt{5}\\\\\Rightarrow \frac{(6+2\sqrt{5}-6+2\sqrt{5})}{4} = a+ b\sqrt{5}\\\\\Rightarrow \frac{4\sqrt{5}}{4} = a+ b\sqrt{5}\\\\\Rightarrow \sqrt{5} = a+ b\sqrt{5}$

$\Rightarrow 0a+ b\sqrt{5} = a+ b\sqrt{5}\\\\\Rightarrow a=0 \ \ \ \ _{and} \ \ \ \ b= 1$

The final value of a and b is 0,1