Question

5+root3/7+2root3=a-broot3

Answer 1

Answer:

$\boxed{\bold{\blue{a = \dfrac{29}{27} \: \: and \: \: b = \dfrac{1}{9}}}}$

Step-by-step explanation:

$\bold{Given} = > \\ \dfrac{5 + \sqrt{3} }{7 + 2 \sqrt{3} } = a - b \sqrt{3}$

$to \: find = > \: value \: of \: a \: and \: b$

$\bold{Concept \: used} = > \\ (a + b) \: (a - b) = {a}^{2} - {b}^{2}$

$\bold{Solution} = > \\ a - b \sqrt{3} = \dfrac{5 + \sqrt{3} }{7 +2 \sqrt{3} }$

$multiplying \: in \: numerator \\\: and \: denomintor \: by \: conjugate \: of \: denominator$

$= \dfrac{(5 + \sqrt{3}) \: (7 - 2 \sqrt{3} ) }{(7 + 2 \sqrt{3} ) \: (7 - 2 \sqrt{3} )}$

$= \dfrac{35 - 10 \sqrt{3} + 7 \sqrt{3} - 6 }{ {(7)}^{2} - {(2 \sqrt{3} )}^{2} }$

$= \dfrac{29 - 3 \sqrt{3} }{49 - 12}$

$= \dfrac{29 - 3 \sqrt{3} }{27}$

$a - b \sqrt{3} = \dfrac{29}{27} - \dfrac{3 \sqrt{3} }{27}$

$= > a - b \sqrt{3} = \dfrac{29}{27} - \dfrac{1}{9} \sqrt{3}$

$on \: comparing \: both \: sides$

$a \: = \: \dfrac{29}{27} \: \: and \: \: b \: = \: \dfrac{1}{9}$