Question

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the green coloured mark question .ok
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Answer 1

Hope u like my process
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Concept of rational numbers
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=> A rational number is any number that has some definite terms and are not extended

for example :

1, 2, 6, 90 ,1.25, 7.8,..

=> An irrational number is any number that has some indefinite terms and are taken as per approximate values..

For example:

The values of e, π,√2...


Given..
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$\bf = > \frac{7 - 4 \sqrt{3} }{7 + 4 \sqrt{3} } = a + b \sqrt{3}$
Now..

Simplifying the LHS term
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$= > \frac{7 - 4 \sqrt{3} }{7 + 4 \sqrt{3} } \\ \\ = > \frac{7 - 4 \sqrt{3} }{7 + 4\sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } \\ \\ = > \frac{ {(7 - 4 \sqrt{3}) }^{2} }{ {7}^{2} - (4 \sqrt{3}) {}^{2} } \\ \\ = > \frac{ {7}^{2} - 2 (7)(4 \sqrt{3}) + {(4 \sqrt{3}) }^{2} }{49 - 48} \\ \\ = > \frac{49 - 56 \sqrt{3} + 48 }{1} \\ \\ = > \bf \underline{ 97 - 56 \sqrt{3} }$

So..

$= > a + b \sqrt{3} = \frac{7 - 4 \sqrt{3} }{7 + 4 \sqrt{3} } = 97 - 56 \sqrt{3} \\ \\ so... \\ \\ = > \boxed{\bf a = \blue{97 }\: \: \: and \: \: b = \blue{ - 56 }}$
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