Question

x=1-√2,find the value of (x-1/x)^3

Answer 1

ʜᴇʟʟᴏ ᴍᴀᴛᴇ!

x = 1 - √2

1/x = 1/( 1 - √2 )

Rationalize

1/( 1 - √2 ) × ( 1 + √2 )/( 1 + √2 )
= ( 1 + √2 )/ - 1 = - 1 - √2

( x - 1/x )³ = [ 1 - √2 - ( - 1 - √2 ) ]³
= [ 1 - √2 + 1 + √2 ]³ = 2³ = 8

Hope it helps

Answer 2

$x = 1 - \sqrt{2 } \\ \frac{1}{x} = \frac{1}{1 - \sqrt{2} } = \frac{1 + \sqrt{2} }{(1 - \sqrt{2} )(1 + \sqrt{2} } \\ = \frac{1 + \sqrt{2} }{ {1}^{2} - {( \sqrt{2}) }^{2} } = \frac{1 + \sqrt{2} }{1 - 2} \\ = \frac{1 + \sqrt{2} }{ - 1} = - (1 + \sqrt{2} ) \\ x - \frac{1}{x} = 1 - \sqrt{2} - ( - 1 - \sqrt{2}) \\ = 1 - \sqrt{2} + 1 + \sqrt{2 } \\ = 2 \\ {(x - \frac{1}{x}) }^{3} = {2}^{3} = 8$