Question

Simplify:-
$\frac{7 + 3 \sqrt{5} }{7 - 3 \sqrt{5} }$
​

Answer 1

Answer:

$\frac{47 - 21 \sqrt{5} }{2}$

Step-by-step explanation:

ATQ,

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

So, we need to rationalize this equation

$\frac{7 + 3 \sqrt{5} }{7 - 3 \sqrt{5} } \times \frac{7 + 3 \sqrt{5} }{7 + 3 \sqrt{5} } \:$

Multiplying both parts of fraction

$\frac{ {(7 + 3 \sqrt{5} )}^{2} }{(7 + 3 \sqrt{5})(7 - 3 \sqrt{5} ) } \:$

Applying the 1st and 3rd identity

$\frac{ {7}^{2} + 2(7 \times 3 \sqrt{5}) + {(3 \sqrt{5} )}^{2} }{ {7}^{2} - {(3 \sqrt{5}) }^{2} }$

Simplifying

$\frac{49 + 42 \sqrt{5} + 45 }{49 - 45}$

Further simplifying

$\frac{94 + 42 \sqrt{5} }{4 }$

Still further simplifying

$\frac{2(47 - 21 \sqrt{5}) }{4}$

Further simplifying, we get

$\frac{47 - 21 \sqrt{5} }{2}$

Please check the answer. I may know the method, but the equations may themselves be incorrect.