Precalculus

Factor and express the answer as a quotient with positive exponents.

x^(-2)-4x^(-5)+3x^(-8)

\(x^{-2}-4x^{-5}+3x^{-8}\\ =x^{-8}(x^6-4x^3+3)\\ =\frac{x^6-4x^3+3}{x^8}\)

Factor and express the answer as a quotient with positive exponents.

x^(-2)-4x^(-5)+3x^(-8)

\(\begin{array}{|rcll|} \hline && x^{-2}-4x^{-5}+3x^{-8} \\ &=& x^{-8}\left( \frac{x^{-2}} {x^{-8}} - \frac{4x^{-5}} {x^{-8}} +3 \frac{ x^{-8} } {x^{-8}} \right) \\ &=& x^{-8} ( x^{-2} x^{8} - 4x^{-5} x^{8} +3 ) \\ &=& x^{-8} ( x^{-2+8} - 4x^{-5+8} +3 \\ &=& x^{-8} ( x^{6} - 4x^{3} +3 ) \\ &=& x^{-8} (x^3-1)(x^3-3) \\ &=& \dfrac{(x^3-1)(x^3-3)}{x^{8}} \\ \hline \end{array}\)

[ ( x^3 - 1) ( x^3 - 3 ) ]           [ ( x - 1) (x^2 + x + 1 ) ( x^3 - 3) ]

_________________   =      __________________________

       x^8                                                  x^8