Question

simplify: (root5 -root3) /(root 5+root3)

Answer 1

$\frac{ (\sqrt{5} - \sqrt{3})( \sqrt{5} - \sqrt{3} )}{( \sqrt{5} + \sqrt{3})( \sqrt{5} - \sqrt{3} ) } \\ = \frac{5 + 3 - 2 \sqrt{15} }{5 - 3} \\ = \frac{8 - 2 \sqrt{5} }{2} \\ = \frac{2(4 - \sqrt{5}) }{2} \\ = 4 - \sqrt{5} \: \: \: \: ans$

Answer 2

Answer:

4 - √15

Step-by-step explanation:

Given:- The given expression is (√5 - √3) / (√5 + √3)

To find:- Value of the given expression.

Solution:-

As we know that √5 and √3 both are irrational number and the difference of irrational numbers is always an irrational number.

We will rationalize the above expression by making the denominator a rational number.

So we will multiply the numerator and denominator by (√5 - √3)

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

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

= $\frac{(\sqrt{5} -\sqrt{3} )^{2} }{(\sqrt{5})^{2}-(\sqrt{3})^{2} }$        [∵ a² - b² = (a + b)(a - b)]

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

= $\frac{(\sqrt{5})^{2}+(\sqrt{3} )^{2} - 2\sqrt{5} \sqrt{3} }{2 }$       [∵ (a - b)² = a² + b² - 2ab]

= $\frac{5+3-2\sqrt{15} }{2 }$

= $\frac{8-2\sqrt{15} }{2 }$

= $\frac{2(4-\sqrt{15}) }{2 }$

= $(4-\sqrt{15})$

Therefore, the value of (√5 - √3) / (√5 + √3) is (4 - √15).

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