Question

find the value of (x-1/x)^3if value of x=1+√2

Answer 1

Hi ,

It is given that ,

x = 1 + √2 ---( 1 )

1/x = 1/( 1 + √2 )

= ( √2 - 1 )/[ ( √2 + 1 )( √2 - 1 )]

= ( √2 - 1 )/[ ( √2 )² - 1² ]

= ( √2 - 1 )/( 2 - 1 )

= √2 - 1 ----( 2 )

x - 1/x = ( 1 ) - ( 2 )

= 1 + √2 - ( √2 - 1 )

= 1 + √2 - √2 + 1

= 2 ---( 3 )

( x - 1/x )³ = 2³ = 8

I hope this helps you.

: )

Answer 2

$\bf \large \it \: Hey \: User!!!$

given x = 1 + √2

it can be written as = √2 + 1

$\sf \:therefore \: \frac{1}{x} = \frac{1}{ \sqrt{2} + 1 } \\ \\ \sf \: = \frac{1}{ \sqrt{2 } + 1} \times \frac{ \sqrt{2} - 1 }{ \sqrt{2} - 1 } \\ \\ \sf = \frac{ \sqrt{2} - 1 }{ {( \sqrt{2} )}^{2} - {( 1) }^{2} } \\ \\ \sf = \frac{ \sqrt{2} - 1 }{2 - 1} \\ \\ \sf = \frac{ \sqrt{2} - 1 }{ 1} = \sqrt{2} - 1$

we have to find the value of (x - 1/x)³

we got the value of 1/x.

let us solve it further...

$\sf > > ({ x - \frac{1}{x} })^{3} \\ \sf ={ ( 1 + \sqrt{2} - ( \sqrt{2} - 1 ))}^{3} \\ \sf = {(1 + \sqrt{2} - \sqrt{2} + 1) }^{3} \\ \sf = {2}^{3} \\ \sf = 2 \times 2 \times 2 \\ \sf = \boxed{8} \: final \: answer$

$\bf \large \it{Cheers!!!}$