2002 Math OSK, Number 5

Algebra Level pending

Suppose that x − n { x }^{ -n } equals to ( 1 x ) n (\frac { 1 }{ x } { ) }^{ n } for every real number x. Then a 3 − a − 3 { a }^{ 3 }-{ a }^{ -3 } equals to...

( a − 1 a ) ( a 2 − 2 + 1 a 2 ) (a-\frac { 1 }{ a } )({ a }^{ 2 }-2+\frac { 1 }{ { a }^{ 2 } } ) ( 1 a − a ) ( a 2 − 1 + 1 a 2 ) (\frac { 1 }{ a } -a)({ a }^{ 2 }-1+\frac { 1 }{ { a }^{ 2 } } ) ( 1 a − a ) ( a 2 + 1 + 1 a 2 ) (\frac { 1 }{ a } -a)({ a }^{ 2 }+1+\frac { 1 }{ { a }^{ 2 } } ) ( a − 1 a ) ( a 2 + 1 + 1 a 2 ) (a-\frac { 1 }{ a } )({ a }^{ 2 }+1+\frac { 1 }{ { a }^{ 2 } } ) None of the other answers are true

Abyoso Hapsoro by Abyoso Hapsoro , 22, Indonesia

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1 solution

Nelson Mandela
Apr 4, 2015

so, a 3 − a − 3 { a }^{ 3 }-{ a }^{ -3 } = a 3 − 1 a 3 { a }^{ 3 }-{ \frac { 1 }{ { a }^{ 3 } } } .

Using a 3 − b 3 = ( a − b ) ( a 2 + b 2 + a b ) { a }^{ 3 }-{ b }^{ 3 }=(a-b)({ a }^{ 2 }+{ b }^{ 2 }+ab) ,

answer is ( a − 1 a ) ( a 2 + 1 a 2 + 1 ) (a-\frac { 1 }{ a } )({ a }^{ 2 }+{ \frac { 1 }{ { a }^{ 2 } } }+1)

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