Question

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Answer 1

Question :

If x + 1/x = 3, calculate x² + 1/x², x³ + 1/x³ and x⁴ + 1/x⁴

Answer :

x + 1/x = 3

Squaring on both sides

( x + 1/x )² = 3²

Using algebraic identity (a + b)² = a² + b² + 2ab

x² + 1/x² + 2( x )( 1/x ) = 9

x² + 1/x² + 2 = 9

x² + 1/x² = 9 - 2

x² + 1/x² = 7

Squaring on both sides

( x² + 1/x² )² = 7²

Using algebraic identity (a + b)² = a² + b² + 2ab

( x² )² + ( 1/x² )² + 2( x² )( 1/x² ) = 49

x⁴ + 1/x⁴ + 2 = 49

x⁴ + 1/x⁴ = 49 - 2

x⁴ + 1/x⁴ = 47

Now find x³ + 1/x³

x + 1/x = 3

Cubing on both sides

( x + 1/x )³ = 3³

Using algebraic identity (a + b)³ = a³ + b³ + 3ab(a + b)

x³ + 1/x³ + 3( x )( 1/x )( x + 1/x ) = 27

x³ + 1/x³ + 3( 3 ) = 27

x³ + 1/x³ + 9 = 27

x³ + 1/x³ = 27 - 9

x³ + 1/x³ = 18