Question

simplify by rationalizing the denominator:√7-√5/√7+√5

Answer 1

here is answer for your question

Answer 2

$\displaystyle \sf{ \frac{ \sqrt{7} - \sqrt{5} }{ \sqrt{7} + \sqrt{5} } } = 6 - \sqrt{35}$

Given :

$\displaystyle \sf{ \frac{ \sqrt{7} - \sqrt{5} }{ \sqrt{7} + \sqrt{5} } }$

To find :

To rationalize the denominator

Solution :

Step 1 of 2 :

Write down the given expression

Here the given expression is

$\displaystyle \sf{ \frac{ \sqrt{7} - \sqrt{5} }{ \sqrt{7} + \sqrt{5} } }$

Step 2 of 2 :

Rationalize the denominator

$\displaystyle \sf{ \frac{ \sqrt{7} - \sqrt{5} }{ \sqrt{7} + \sqrt{5} } }$

Multiplying both of the numerator and denominator by √7 - √5 we get

$\displaystyle \sf{ \frac{ \sqrt{7} - \sqrt{5} }{ \sqrt{7} + \sqrt{5} } }$

$\displaystyle \sf{ = \frac{( \sqrt{7} - \sqrt{5})( \sqrt{7} - \sqrt{5} )}{ (\sqrt{7} + \sqrt{5})(\sqrt{7} - \sqrt{5} ) } }$

$\displaystyle \sf{ = \frac{{( \sqrt{7} - \sqrt{5}) }^{2} }{ {(\sqrt{7} )}^{2} - {( \sqrt{5} )}^{2} } }$

$\displaystyle \sf{ = \frac{ {(\sqrt{7} )}^{2} - 2 \times \sqrt{7} \times \sqrt{5} + {( \sqrt{5} )}^{2} }{ 7 - 5} }$

$\displaystyle \sf{ = \frac{ 7 - 2 \sqrt{35} + 5 }{ 2} }$

$\displaystyle \sf{ = \frac{ 12 - 2 \sqrt{35} }{ 2} }$

$\displaystyle \sf{ = \frac{ 2(6 - \sqrt{35} ) }{ 2} }$

$\displaystyle \sf{ = 6 - \sqrt{35} }$

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