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Answer 1

Question :

$\boldsymbol{ \dfrac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3}} = a +b \sqrt{3}}$

To find

• The value of a and b

Solution

Firstly, we will rationalize the denominator.

Multiply the denominator and numerator by the same number.

$\boldsymbol{ \dfrac{(5 + 2 \sqrt{3})(7 - 4 \sqrt{3} ) }{(7 + 4 \sqrt{3})(7 - 4 \sqrt{3} )}}$

Here, we will use an algebraic identity.

Identity to be used :-

$\gray{\pmb{(a - b)(a + b) = {a}^{2} - {b}^{2}}}$

$\\ : \implies \boldsymbol{ \dfrac{(5 + 2 \sqrt{3})(7 - 4 \sqrt{3} ) }{(7)^{2} - (4 \sqrt{3})^{2}}}$

$\\ : \implies \boldsymbol{ \dfrac{(5 + 2 \sqrt{3})(7 - 4 \sqrt{3} ) }{49 - 48}}$

[Note :- When two underoots are multiplied, underoot gets removed.]

$\gray{\pmb{ \sqrt{3} \times \sqrt{3} = 3 }}$

$\\ : \implies \boldsymbol{ \dfrac{(5 + 2 \sqrt{3})(7 - 4 \sqrt{3} ) }{1}}$

$\\ : \implies \boldsymbol{ \dfrac{5 (7 - 4 \sqrt{3} ) + 2 \sqrt{3} (7 - 4 \sqrt{3} ) }{1}}$

$\\ : \implies \boldsymbol{35 - 20 \sqrt{3} + 28 \sqrt{3} - 24}$

$\\ : \implies \boldsymbol{11 + 8 \sqrt{3}}$

$\\ : \implies \boldsymbol{11 + 8 \sqrt{3} = a + b \sqrt{3} }$

$\\ : \implies \boldsymbol{a = 11 }$

$\\ : \implies \boldsymbol{b \sqrt{3} = 8 \sqrt{3} }$

$\\ : \implies \boldsymbol{b \cancel{\sqrt{3}}= 8 \cancel{\sqrt{3} } }$

$\\ : \implies \boldsymbol{b =8 }$

$\therefore \underline{ \orange{\pmb{value \: of \: a = 11 \: and \: b = 8}}}$

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Some useful identities :-

• (a + b)² = a² + b² + 2ab
• (a - b)² = a² + b² - 2ab
• (a + b)³ = a³ + b³ + 3ab(a + b)
• (a - b)³ = a³ - b³ - 3ab(a - b)