Question

8. If x=2+√13,find (x+1/x)^3​

Answer 1

Answer:

Required value of ( x + 1 / x )^3 is ( 66496 + 20680√13 ) / 729.

Step-by-step explanation:

Given,

x = 2 + √13

1 / x = 1 / ( 2 + √13 ) { Reciprocal of x }

$\implies \dfrac{1}{x}=\dfrac{1}{2}+\sqrt{13}$

Using Rationalisation : Multiply and divide by the original denominator with opposite signs between the rational and irrational number so that the result will be rational number.

So, here, we have to multiply and divide by 2 - √13.

$\implies \dfrac{1}{x}=\dfrac{1}{2+\sqrt{13}}\times\dfrac{2-\sqrt{13}}{2-\sqrt{13}}\\\\\\\implies \dfrac{1}{x}=\dfrac{2-\sqrt{13}}{(2+\sqrt{13})(2-\sqrt{13})}$

From the properties of explanation :

• ( a + b )( a - b ) = a^2 - b^2

$\implies \dfrac{1}{x}=\dfrac{2-\sqrt{13}}{(2) ^2-(\sqrt{13})^2}\\\\\\\impliee\dfrac{1}{x}=\dfrac{2-\sqrt{13}}{4-13}\\\\\\\implies\dfrac{1}{x}=\dfrac{2-\sqrt{13}}{-9}\\\\\\\implies\dfrac{1}{x}=\dfrac{\sqrt{13}-2}{9}$

Therefore,

= > ( x + 1 / x )^3

= > [ ( 2 + √13 ) + ( √13 - 2 ) / 9 ]^3

= > [ { 9( 2 + √13 ) + √13 - 2 } / 9 ]^3

= > [ ( 18 + 9√13 + √13 - 2 ) / 9 ]^3

= > [ ( 16 + 10√13 ) / 9 ]^3

= > ( 66496 + 20680√13 ) / 729

Hence the required value of ( x + 1 / x )^3 is ( 66496 + 20680√13 ) / 729.