Question

if x⁴+1/x⁴ is 194.find the value of x³+1/x³.

Answer 1

Hi ,

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We know the algebraic identity ,

a² + b² + 2ab = ( a + b )²

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Now ,

x⁴ + 1/x⁴ = 194 ( given )---( 1 )

( x² )² + ( 1/x² )² + 2 × x² + 1/x² = 194 + 2

( x ² + 1/x² )² = 196

( x² + 1/x² )² = ( 14 )²

x² + 1/x² = 14 ---( 2 )

x² + 1/x² + 2 = 14 + 2

( x + 1/x )² = 16

( x + 1/x )² = 4²

x + 1/x = 4 ----( 3 )

Now ,

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We know the algebraic identity ,

a³ + b³ + 3ab( a + b ) = ( a + b )³

Or

a³ + b³ = ( a + b )³ - 3ab( a + b )

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Now ,

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

= 4³ - 3 × 4

= 64 - 12

= 52

I hope this helps you.

: )