Express $1/(1+2(^3√7)-5(^3√49))$ in the form $a+b(^3√7)+c(^3√49)$
I have a couple ideas, but I may be wrong:
I want to compute the norm of the given expression. My result is $x^3 - 3x^2 + 33x +12,191$, but that's not what I want.
I want to compute the given expression by taking the conjugate of the denominator, namely $(1+2(^3√7)-5(^3√49)$. However, I do not know how I could do it with a trinomial. If it was a binomial, I would have no problem. Any suggestions?
there is a conjugate available, it amounts to factoring $x^3 + y^3 + z^3 - 3xyz$ but throwing in coefficients.
Take $ t^3 = d ,$ where your $d=7,$ anyway the cube root will be called $t.$
$$ \left( x+ty+t^2 z \right) \left( x^2 + t^2 y^2 + t^4 z^2 - t^3 yz - t^2 z x - t x y \right) $$ $$ = x^3 + t^3 y^3 + t^6 z^3 - 3 t^3 xyz$$ $$ = x^3 + d y^3 + d^2 z^3 - 3dxyz $$