Question

Simplify by rationalizing the denominator: (7 + 3√5)/(3+ √5) - (7 - 3√5)/(3 - √5) ​

Answer 1

Answer:

please consider this answer

Answer 2

Answer:

$\frac{(7 + 3\sqrt{5}) }{(3+ \sqrt{5}) } - \frac{(7 - 3\sqrt{5}) }{(3- \sqrt{5}) }$

I. $\frac{(7 + 3\sqrt{5}) }{(3+ \sqrt{5}) }$

Rationalizing factor = $3 -\sqrt{5}$

$\frac{(7 + 3\sqrt{5}) }{(3+ \sqrt{5}) }$  X  $\frac{(3 -\sqrt{5})}{(3 -\sqrt{5})}$

$= \frac{21 - 7 \sqrt{5} + 9\sqrt{5} - (3 X 5)}{(3^{2}) - (\sqrt{5} ^{2}) }$

$= \frac{21 - 7 \sqrt{5} + 9\sqrt{5} - 15}{ 9 - 5 }$

$= \frac{6 + 2\sqrt{5} }{4 }$

II. $\frac{(7 - 3\sqrt{5}) }{(3- \sqrt{5}) }$

Rationalizing factor = $3 + \sqrt{5}$

$\frac{(7 - 3\sqrt{5}) }{(3- \sqrt{5}) }$  X $\frac{(3 + \sqrt{5})}{(3 + \sqrt{5})}$

$= \frac{21 + 7 \sqrt{5} - 9\sqrt{5} - (3 X 5)}{(3^{2}) - (\sqrt{5} ^{2}) }$

$= \frac{21 + 7 \sqrt{5} - 9\sqrt{5} - 15}{ 9 - 5 }$

$= \frac{6 - 2\sqrt{5} }{4 }$

I - II :

$= \frac{6 + 2\sqrt{5} }{4 } - \frac{6 - 2\sqrt{5} }{4 }$

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

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