Question

If x=1-√2,find the value of (x-1/x)^4

Answer 1

16

Step-by-step explanation:

Given:

x = 1 - √2

To find:

$\text{\tt{The value of}} \: \: {(x - \dfrac{1}{x})}^{4}$

Answer in 3 steps:

1. Find the value of 1/x.

2. Find the value of (x - 1/x)

3. Simplify and Answer,

Solution:

$x = 1 - \sqrt{2} \\ \\ \frac{1}{x} = \frac{1}{1 - \sqrt{2}}$

[Rationalizing the Denominator]

$= \frac{1}{(1 - \sqrt{2})} \times \frac{(1 + \sqrt{2}) }{( 1+ \sqrt{2} )} \\ \\ = \frac{1 + \sqrt{2} }{ {(1)}^{2} - {(\sqrt{2})}^{2} } [\because {a}^{2} - {b}^{2} = (a + b)(a - b)] \\ \\ = \frac{1 + \sqrt{2} }{1 - 2} \\ \\ = \frac{1 + \sqrt{2} }{ - 1} \\ \\ \therefore \: \: \frac{1}{x} = (- 1 - \sqrt{2})$

Again,

$x - \frac{1}{x} \\ \\ = 1 - \sqrt{2} - ( - 1 - \sqrt{2}) \\ \\ = 1 - \sqrt{2} + 1 + \sqrt{2}\\ \\ = 2 \\ \\ \therefore \: (x - \frac{1}{x} ) = 2 \\ \\ Or \: \: \: (x - \frac{1}{x} ) {}^{4} = {(2)}^{4} \\ \\ Or \: \: \: (x - \frac{1}{x} ) {}^{4} = 16$

Therefore, Required answer is 16.

Answer 2

16

Step-by-step explanation:

Given:-

$x = (1 - \sqrt{2} ) \\ \frac{1}{x} = \frac{1}{1 - \sqrt{2} } = ( \frac{1}{1 - \sqrt{2} } \times \frac{1 + \sqrt{2} }{1 + \sqrt{2} } ) = \frac{1 + \sqrt{2} }{ - 1} =( -1 - \sqrt{2} )$

Solution:-

${(x - \frac{1}{x} )}^{4} \\ \\ = {(1 - \sqrt{2} -( - 1 - \sqrt{2} ))}^{4} \\ \\ = {(1 - \sqrt{2} + 1 + \sqrt{2} )}^{4} \\ \\ = {(2}^{4} \\ \\ = 16$