Given
x = ( √3 + √2 ) / ( √3 - √2 )
Rationalising the denominator by multiplying by its conjugate √3 + √2
x = ( √3 + √2 )² / [ (√3 - √2 )( √3 + √2 ) ]
Using algebraic identities ( x + y )² = x² + y² + 2xy and ( x + y)( x - y) = x² - y²
x = [ (√3)² + (√2)² + 2√6 ] / [ (√3)² - (√2)² ]
x = [ 3 + 2 + 2√6 ] / ( 3 - 2 )
x = ( 5 + 2√6 ) / 1
x = 5 + 2√6
Now find x⁴
x⁴ = ( 5 + 2√6 )⁴
= [ ( 5 + 2√6 )² ]²
= [ 5² + (2√6)² + 20√6 ]²
= ( 25 + 24 + 20√6 )²
= ( 49 + 20√6 )²
= 49² + (20√6)² + 196√6
= 4801 + 1960√6
Now find 1/x⁴
1/x⁴ = 1/( 4801 + 1960√6 )
Rationalising the denominator by multiplying by its conjugate 4801 - 1960√6
= ( 4801 - 1960√6 ) / [ ( 4801 + 1960√6 )( 4801 - 1960√6 ) ]
Using algebraic identity ( x + y) ( x - y) = x² - y²
= ( 4801 - 1960√6 ) / [ 4801² - ( 1960√6 )² ]
= ( 4801 - 1960√6 ) / ( 23049601 - 23049600 )
= ( 4801 - 1960√6 ) / 1
= 4801 - 1960√6
Now find x⁴ + 1/x⁴
x⁴ + 1/x⁴ = 4801 + 1960√6 + 4801 - 1960√6
= 9602
Therefore the value of x⁴ + 1/x⁴ is 9602.
