Question

simplify the following ​

Answer 1

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

$⇒( {3)}^{2} - {( \sqrt{5})}^{2}$

Taking the LCM = Multiplying the denominator

Using the identity of a²-b² = (a+b)(a-b)

$⇒( {3})^{2} - ( { \sqrt{5} })^{2}$

$⇒ 9 - 5$

$⇒4$

A/q ,

$\frac{(7 + 3 \sqrt{5}) \times (3 - \sqrt{5} ) }{4} - \frac{(7 - 3 \sqrt{5} )(3 + \sqrt{5}) }{4}$

$⇒\frac{7(3 - \sqrt{5}) + 3 \sqrt{5} (3 - \sqrt{5} ) }{4} - \frac{7(3 + \sqrt{5}) - 3 \sqrt{5} (3 + \sqrt{5} ) }{4}$

$⇒ \frac{21 - 7 \sqrt{5} + 9 \sqrt{5} - (3 \times 5) }{4} - \frac{21 + 7 \sqrt{5} - 9 \sqrt{5} - (3 \times 5) }{4}$

$⇒ \frac{21 - 7 \sqrt{5} + 9 \sqrt{5} - 15 }{4} - \frac{21 + 7 \sqrt{5} - 9 \sqrt{5} - 15}{4} </p><p>$

$⇒ \frac{21 - 7 \sqrt{5} + 9 \sqrt{5} - 15 - 21 - 7 \sqrt{5} + 9 \sqrt{5} + 15 }{4}$

$⇒\frac{21 - 21 + 15 - 15 - 7 \sqrt{5} - 7 \sqrt{5} + 9 \sqrt{5} + 9 \sqrt{5} }{4}$

$⇒ \frac{ - 7 \sqrt{5} - 7 \sqrt{5} + 9 \sqrt{5} + 9 \sqrt{5} }{4}$

$⇒ \frac{ 18 \sqrt{5} - 14 \sqrt{5} }{4}$

$now \: taking \: \sqrt{5} \: as \: common$

$⇒ \frac{\sqrt{5} \: (18 - 14)}{4}$

$⇒ \frac{4 \sqrt{5} }{4}$

$⇒ \sqrt{5}$

$\pink{\large\mathfrak {Hope \: it \: helps \: you}}$