Question

If x= 9-4 root 5 find x power 6 + 1/x power 6

Answer 1

Answer:

Required value of x^6 + 1 / x^6 is 33385282.

Step-by-step-explanation:

Given,

x = 9 - 4√5

= > x = 9 - 4√5

= > 1 / x = 1 / ( 9 - 4√5 )

Using Rationalisation : In right hand side , multiplying and dividing the original numerator and denominator by the original denominator but with the opposite signs between the rational and irrational terms.

It means : Here, we have to multiply by 9 + 4√5.

Therefore,

$\implies \dfrac{1}{x}=\dfrac{1}{9-4\sqrt5} \times \dfrac{9+4\sqrt5}{9+4\sqrt5}\\\\\\\implies \dfrac{1}{x}=\dfrac{9+4\sqrt5}{(9-4\sqrt5)(9+4\sqrt5)}$

From the properties of expansion :

( a + b )( a - b ) = a^2 - b^2

$\implies \dfrac{1}{x} =\dfrac{9+4\sqrt5}{(9)^2-(4\sqrt5)^2}\\\\\\\implies \dfrac{1}{x}=\dfrac{9+4\sqrt5}{81-80}\\\\\\\implies \dfrac{1}{x}=9+4\sqrt5$

Then,

= > x^6 + 1 / x^6

= > ( x^2 )^3 + ( 1 / x^2 )^3

= > ( x^2 + 1 / x^2 )( x^4 - x^2 / x^2 + 1 / x^4 ) { a^3 + b^3 = ( a + b )( a^2 - ab + b^2 ) }

= > ( x^2 + 1 / x^2 )( x^4 - 1 + 1 / x^4 )

= > { ( x + 1 / x )^2 - 2( x × 1 / x ) }{ x^4 + 1 / x^4 - 1 } { ( a + b )^2 - 2ab = a^2 + b^2 }

= > { ( x + 1 / x )^2 - 2 }{ x^4 + 1 / x^4 - 1 }

= > { ( 9 - 4√5 + 9 + 4√5 )^2 - 2 } { ( x^2 )^2 + ( 1 / x^2 )^2 - 1 }

= > { ( 18 )^2 - 2 }[ { ( x^2 + 1 / x^2 )^2 - 2 - 1 } { ( x^2 + 1 / x^2 )^2 - 2 = x^4 + 1 / x^4 }

= > ( 324 - 2 )[ { ( x + 1 / x )^2 - 2 }^2 - 3 }

= > ( 324 - 2 ){ { ( 9 - 4√5 + 9 + 4√5 )^2 - 2 }^2 - 3 }

= > ( 322 ){ ( 18^2 - 2 )^2 - 3 }

= > ( 322 )( ( 324 - 2 )^2 - 3 )

= > 322 ×( ( 322 )^2 - 3 )

= > 322 x 103681

= > 33,385,282

Hence the required value of x^6 + 1 / x^6 is 33,385,282.