Question

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Answer 1

Question:

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

Answer:

Rationalize the denominator

$\frac{7 + \sqrt{5} }{7 - \sqrt{5} } \times \frac{7 + \sqrt{5} }{7 + \sqrt{5} }$

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

$\frac{49 + 5 + 2 \times 7 \times \sqrt{5} }{49 - 5}$

$\frac{54 + 14 \sqrt{5} }{44}$

$\frac{54}{44} + \frac{14 \sqrt{5} }{44}$

$a = \frac{54}{44} = \frac{27}{22}$

$b = \frac{14 \sqrt{5} }{44} = \frac{7 \sqrt{5} }{22}$