Question

Find the value of a and b such that (5 + sqrt(3))/(7 + 2sqrt(3)) = a - b * sqrt(3)​

Answer 1

ANSWER:

$a = \frac{29}{37}$

$b = \frac{3}{37}$

EXPLANATION:

$\frac{5 + \sqrt{3} }{7 + 2 \sqrt{3} } = a - b \sqrt{3}$

$\frac{5 + \sqrt{3} }{7 + 2 \sqrt{3} } \times \frac{7 - 2 \sqrt{3} }{7 - 2 \sqrt{3} } = a - b \sqrt{3}$

$\frac{(5 + \sqrt{3}) (7 - 2 \sqrt{3})}{(7 + 2 \sqrt{3})(7 - 2 \sqrt{3}) } = a - b \sqrt{3}$

$\frac{7(5 + \sqrt{3} ) - 2 \sqrt{3}(5 + \sqrt{3}) }{ {(7)}^{2} - {(2 \sqrt{3}) }^{2} } = a - b \sqrt{3}$

$\frac{35 + 7 \sqrt{3} - 10 \sqrt{3} - 2( \sqrt{3})^{2} }{49 - 4(3)} = a - b \sqrt{3}$

$\frac{35 - 3 \sqrt{3} - 2({3}) }{49 - 12} = a - b \sqrt{3}$

$\frac{35 - 3 \sqrt{3} - 6 }{37} = a - b \sqrt{3}$

$\frac{35 - 6 - 3 \sqrt{3} }{37} = a - b \sqrt{3}$

$\frac{29 - 3 \sqrt{ 3} }{37} = a - b \sqrt{3}$

$\frac{29}{37} - \frac{3 \sqrt{3} }{37} = a - b \sqrt{3}$

$\frac{29}{37} - \frac{3}{37} \sqrt{3} = a - b \sqrt{3}$

$comparing \: both \: equations$

$a = \frac{29}{37}$

$b = \frac{3}{37}$

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Answer 2

Answer:

Step-by-step explanation: