Question

if a=9+4root 5 find (a)a+1/a (b)a-1/a​

Answer 1

Given:

$a = 9 + 4 \sqrt{5}$

To find:

The value of

$\: a) \: a + \frac{1}{a} \\ \\ b) \: a - \frac{1}{a}$

Solution:

$a = 9 + 4 \sqrt{5} \\ \\ = > \frac{1}{a} = \frac{1}{9 + 4 \sqrt{5} }$

Rationalizing the denominator.

$= > \frac{1}{9 + 4 \sqrt{5} } \times \frac{9 - 4 \sqrt{5} }{9 - 4 \sqrt{5} } \\ \\ = > \frac{9 - 4 \sqrt{5} }{(9) {}^{2} - (4 \sqrt{5}) {}^{2} }$

We know that (a+b)(a-b)= a²-b²

$= > \frac{9 - 4 \sqrt{5} }{81 - 80} \\ \\ = > \frac{1}{a} = 9 - 4 \sqrt{5}$

So, we will put the values of a and 1/a

a)

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

b)

$a - \frac{1}{a} \\ \\ = 9 + 4 \sqrt{5} - (9 - 4 \sqrt{5} ) \\ \\ = 9 + 4 \sqrt{5} - 9 + 4 \sqrt{5} \\ \\ = \cancel{9} + 4 \sqrt{5} - \cancel{9} + 4 \sqrt{5 } \\ \\ = 4 \sqrt{5} + 4 \sqrt{5} \\ \\ = 8 \sqrt{5}$

Hope it helps you

Mark as brainliest