Question

rationalise 4+ root 5 + 4 - root 5/4- root 5+4+root 5

Answer 1

hope you understand the answer

Answer 2

$\displaystyle \sf{ \frac{4 + \sqrt{5} }{4 - \sqrt{5} } + \frac{4 - \sqrt{5} }{4 + \sqrt{5} } } = \frac{42}{11}$

Given :

$\displaystyle \sf{ \frac{4 + \sqrt{5} }{4 - \sqrt{5} } + \frac{4 - \sqrt{5} }{4 + \sqrt{5} } }$

To find :

To simplify the expression

Solution :

Step 1 of 2 :

Write down the given expression

The given expression is

$\displaystyle \sf{ \frac{4 + \sqrt{5} }{4 - \sqrt{5} } + \frac{4 - \sqrt{5} }{4 + \sqrt{5} } }$

Step 2 of 2 :

Simplify the expression

$\displaystyle \sf{ \frac{4 + \sqrt{5} }{4 - \sqrt{5} } + \frac{4 - \sqrt{5} }{4 + \sqrt{5} } }$

$\displaystyle \sf = \frac{(4 + \sqrt{5})(4 + \sqrt{5}) }{(4 + \sqrt{5} )(4 - \sqrt{5}) } + \frac{(4 - \sqrt{5})(4 - \sqrt{5} )}{(4 + \sqrt{5} )(4 - \sqrt{5} )}$

$\displaystyle \sf = \frac{{(4 + \sqrt{5})}^{2} }{ {(4)}^{2} - {( \sqrt{5} )}^{2} } + \frac{{(4 - \sqrt{5})}^{2} }{ {(4)}^{2} - {( \sqrt{5} )}^{2} }$

$\displaystyle \sf = \frac{{ {(4)}^{2} + ( \sqrt{5})}^{2} - 2 \times 4 \times \sqrt{5} }{ 16 - 5 } + \frac{{ {(4)}^{2} + ( \sqrt{5})}^{2} + 2 \times 4 \times \sqrt{5} }{ 16 - 5 }$

$\displaystyle \sf = \frac{16 + 5 + 8 \sqrt{5} }{11} + \frac{16 + 5 - 8 \sqrt{5} }{11}$

$\displaystyle \sf = \frac{21+ 8 \sqrt{5} }{11} + \frac{21 - 8 \sqrt{5} }{11}$

$\displaystyle \sf = \frac{21 + 8 \sqrt{5} + 21 - 8 \sqrt{5} }{11}$

$\displaystyle \sf = \frac{42 }{11}$

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