Question

3. Are 7+√5 and 7-√5 rationals? Show whether their sum and product are rational or irrational.

Answer 1

Answer:

$no \: \: \: 7+\sqrt{5} \: and \: 7- \sqrt{5} \: are \: not \: rational \: numbers$

Even they are irrational their product and sum are rational.

Sum:

$(7 + \sqrt{5}) + (7 - \sqrt{5}) = 14$

Here 14 is a rational number.

Product :

$(7 + \sqrt{5} )(7 - \sqrt{5} ) = {7}^{2} - { \sqrt{5} }^{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 49 - 5$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = 44$

Here 44 is a rational number.

Hence the law, the sum and product of two irrational numbers are rational is proved.

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