Question

if X = 3- 2 root 2, then the value of (x-1/x)raised to power 2​

Answer 1

Answer:

Required value of ( x - 1 / x )^2 is 32.

Step-by-step explanation:

Given numeric value of x is 3 - 2√2.

$\implies x=3-2\sqrt{2} \\\\\implies \dfrac{1}{x}=\dfrac{1}{3-2\sqrt{2}}$

Using Rationalization : Multiplying and dividing the right hand side by the original denominator with opposite signs between the rational & irrational numbers.

$\implies \dfrac{1}{x}=\dfrac{1}{3-2\sqrt{2}}\times\dfrac{3+2\sqrt2}{3+2\sqrt2}\\\\\\\implies \dfrac{1}{x}=\dfrac{3+2\sqrt2}{(3-2\sqrt2)(3+2\sqrt2)}$

From the properties of factorisation :

• ( a - b )( a + b ) = a^2 - b^2

$\implies \dfrac{1}{x}=\dfrac{3+2\sqrt2}{(3)^2-(2\sqrt2)^2}\\\\\\\implies \dfrac{1}{x}=\dfrac{3+2\sqrt2}{9-8} = 3+2\sqrt2$

Therefore,

$\implies x - \dfrac{1}{x}= 3 - 2\sqrt2 - 3 - 2\sqrt2 = - 4\sqrt2$

Hence,

= > ( x - 1 / x )^2 = ( - 4√2 )^2

= > ( x - 1 / x )^2 = 32

Hence the required value of ( x - 1 / x )^2 is 32.