Given
x² + 1/x² = 51
Subtracting 2 on both sides
x² + 1/x² - 2 = 51 - 2
It can be written as
x² + ( 1/x )² - 2( x)( 1/x ) = 49
Using algebraic identity a² + b² - 2ab = ( a - b)²
( x - 1/x )² = 49
Taking square root on both sides
x - 1/x = √49 = 7
x - 1/x = 7
Now find x³ - 1/x³
Using algebraic identity a³ - b³ = ( a - b)( a² + b² - ab)
x³ - 1/x³ = ( x - 1/x ) ( x² + 1/x² + x( 1/x ) )
x³ - 1/x³ = 7( 51 + 1 )
x³ - 1/x³ = 52 × 7
x³ - 1/x³ = 364
Therefore the value of x³ - 1/x³ is 364.