Question

if
$x - \frac{1}{x} = 9$
then find the value of ( i )
${x}^{2} + \frac{1}{x {}^{2} }$
( ii )
${x}^{4} + \frac{1}{x {}^{4} }$
​

Answer 1

Given:-

$=> (x - \frac{1}{x}) = 9$

To find:-

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

$=> ({x}^{4} + \frac{1}{ {x}^{4} })$

Solution:-

By squaring both the sides in the equation (x - 1/x)=9, we get:-

$= > (x - \frac{1}{x} ) ^{2} = (9)^{2}$

We know that:-

=>(a-b)² = a² + b² - 2ab

$= > (x) ^{2} + ({ \frac{1}{x} )}^{2} - 2 = 81$

$= > {x}^{2} + \frac{1}{ {x}^{2} } - 2 = 81$

$= > {x}^{2} + \frac{1}{ {x}^{2}} = 81 + 2$

$= > {x}^{2} + \frac{1}{ {x}^{2} } = 83$

Hence, we have got the value of

(x²+1/x²) as 83.

_____________________________________

Now,by squaring both the sides in the equation (x² + 1/x²)=83, we get:-

$= > ( {x}^{2} + \frac{1}{ {x}^{2} } ) ^{2} = (83)^{2}$

We know that:-

=>(a+b)² = a² + b² + 2ab

$= >({ {x}^{2} })^{2} + (\frac{1}{ {x}^{2}})^{2} + 2= 6889$

$= > {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 6889$

$= > {x}^{4} + \frac{1}{ {x}^{4} } = 6889 - 2$

$= > {x}^{4} + \frac{1}{ {x}^{4} } = 6887$

Thus:-

•Value of (x²+1/x²) is 83.

•Value of (x⁴+1/x⁴) is 6887.

Answer 2

Given:-

$=> (x - \frac{1}{x}) = 9$

To find:-

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

$=> ({x}^{4} + \frac{1}{ {x}^{4} })$

Solution:-

By squaring both the sides in the equation (x - 1/x)=9, we get:-

$= > (x - \frac{1}{x} ) ^{2} = (9)^{2}$

We know that:-

=>(a-b)² = a² + b² - 2ab

$= > (x) ^{2} + ({ \frac{1}{x} )}^{2} - 2 = 81$

$= > {x}^{2} + \frac{1}{ {x}^{2} } - 2 = 81$

$= > {x}^{2} + \frac{1}{ {x}^{2}} = 81 + 2$

$= > {x}^{2} + \frac{1}{ {x}^{2} } = 83$

Hence, we have got the value of

(x²+1/x²) as 83.

_____________________________________

Now,by squaring both the sides in the equation (x² + 1/x²)=83, we get:-

$= > ( {x}^{2} + \frac{1}{ {x}^{2} } ) ^{2} = (83)^{2}$

We know that:-

=>(a+b)² = a² + b² + 2ab

$= >({ {x}^{2} })^{2} + (\frac{1}{ {x}^{2}})^{2} + 2= 6889$

$= > {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 6889$

$= > {x}^{4} + \frac{1}{ {x}^{4} } = 6889 - 2$

$= > {x}^{4} + \frac{1}{ {x}^{4} } = 6887$

Thus:-

•Value of (x²+1/x²) is 83.

•Value of (x⁴+1/x⁴) is 6887.

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