Question

find the answer of the following questions ​

Answer 1

Given:

$a + \frac{1}{a } = 6$

To Find:

$(i) \: \: a - \frac{1}{a}$

$(ii) \: \: {a}^{2} - \frac{1}{ {a}^{2} }$

Answer:

First find a² + 1/a²

${a} + \frac{1}{a} = 6$

${(a + \frac{1}{a} )}^{2} = {a}^{2} + \frac{1}{ {a}^{2} } + 2 a \frac{1}{a}$

${6}^{2} =36 = {a}^{2} + \frac{1}{ {a}^{2} } + 2$

${a}^{2} + \frac{1}{ {a}^{2} } = 34$

Now find (i)

${(a - \frac{1}{a} )}^{2} = {a}^{2} + \frac{1}{ {a}^{2} } - 2$

${(a - \frac{1}{a}) }^{2} = 34 - 2 = 32$

$(a - \frac{1}{a }) = \sqrt{32} = 4 \sqrt{2}$

Now find (ii)

$( {a}^{2} - \frac{1}{ {a}^{2} } ) = (a + \frac{1}{a} )(a - \frac{1}{a} )$

$({a}^{2} - \frac{1}{ {a}^{2} } ) = 6 \times 4 \sqrt{2}$

$( {a}^{2} - \frac{1}{ {a}^{2} } ) = 24 \sqrt{2}$

Ans. (i) = 4√2 and (ii) = 24√2