Question

A is equal to whole root 9-4root 5 find the value of root a - 1upon root a

Answer 1

$a = 9 - 4 \sqrt{5} \\ \frac{1}{a} = \frac{1}{ 9 - 4 \sqrt{5} } = \frac{ 9 + 4 \sqrt{5} }{( 9 - 4 \sqrt{5} )( 9 + 4 \sqrt{5} )} \\ = \frac{ 9 + 4 \sqrt{5} }{81 - 80} = 9 + 4 \sqrt{5}$

$a + \frac{1}{a} = ( 9 - 4 \sqrt{5} ) + ( 9 + 4 \sqrt{5} ) \\ = 18$

${( \sqrt{a} - \frac{1}{ \sqrt{a} }) }^{2} = a + \frac{1}{a} - 2 \\ = 18 - 2 = 16 \\ therefore \\ \sqrt{a} - \frac{1}{ \sqrt{a} } = \sqrt{16} = + \: or \: - 4$