Question

if a=5+2√6 , then find the value of a2+1/a2

Answer 1

Identities used :

${(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ \\ {(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab \\ \\ (a + b)(a - b) = {a}^{2} - {b}^{2}$


Given,

a = 5+ 2√6

So,

$\frac{1}{a} = \frac{1}{5 + 2 \sqrt{6} }$

On rationalizing the denominator we get,

$\frac{1}{a} = \frac{1}{5 + 2 \sqrt{6} } \times \frac{5 - 2 \sqrt{6} }{5 - 2 \sqrt{6} } \\ \\ \frac{1}{a} = \frac{5 - 2 \sqrt{6} }{ {(5)}^{2} - {(2 \sqrt{6} )}^{2} } \\ \\ \frac{1}{a} = \frac{5 - 2 \sqrt{6} }{25 - 24} \\ \\ \frac{1}{a} = 5 - 2 \sqrt{6}$

Now,

$a + \frac{1}{a} = (5 + 2 \sqrt{6} ) + (5 - 2 \sqrt{6} ) \\ \\ a + \frac{1}{a} = 5 + 2 \sqrt{6} + 5 - 2 \sqrt{6} \\ \\ a + \frac{1}{a} = 10$

Squaring both the sides we get,

${(a + \frac{1}{a} )}^{2} = {(10)}^{2} \\ \\ {a}^{2} + {( \frac{1}{a} )}^{2} + 2 \times a \times \frac{1}{a} = 100 \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } = 100 - 2 \\ \\ { a }^{2} + \frac{1}{ {a}^{2} } = 98$

Answer 2

Answer:

(

(−3)

5

)×(

(−2)

6

) ×(

(6)

5

)

(2 )