Question

(p - 1/p) (p + 1/p) (p²+ 1/p²) ( p⁴+ 1/p⁴)

Answer 1

Question -

Expand -

$\sf{ ( p - \dfrac{1}{p} ) ( p + \dfrac{1}{p} ) ( p^2 + \dfrac{1}{ p^2 } )( p^4 + \dfrac{1}{ p^4 } ) }$

Solution -

$\sf{ ( p - \dfrac{1}{p} ) ( p + \dfrac{1}{p} ) ( p^2 + \dfrac{1}{ p^2 } )( p^4 + \dfrac{1}{ p^4 } ) } \\ \\ \sf{ => ( p^2 - \dfrac{1}{ p^2 } )( p^2 + \dfrac{1}{ p^2 } )( p^4 + \dfrac{1}{ p^4 } ) } \\ \\ \sf{ => ( p^4 - \dfrac{1}{ p^4} )( p^4 + \dfrac{1}{ p^4 } ) } \\ \\ \sf{ => (p^8 - \dfrac{1}{ p^8 } ) } \\ \\ \sf{ \bold{ Hence \: Expanded } }$

Additional Identities -

$\sf{ (a - b )( a + b ) = a^2 - b^2 } \\ \\ \sf{ ( a + b )^2 = a^2 + 2ab + b^2 } \\ \\ \sf{ ( a - b )^2 = a^2 - 2ab + b^2 }$