[Newbie:] How to discard higher orders of variables?
Subject: [Newbie:] How to discard higher orders of variables?
From: Stavros Macrakis
Date: Thu, 12 Jul 2007 09:14:41 -0400
On 7/11/07, Daniel Lakeland <dlakelan at street-artists.org> wrote:
>
> ...the equation is an 8th order
> polynomial, so it should have 8 solutions. I'm impressed that maxima
> can find them all, given the general nontractability of 5th and higher
> order polynomials.
As you say, the *general* case is intractable. But there are many special
cases which Maxima handles by factoring, decomposing into a composition of
polynomials (polydecomp), etc. So, for example, Maxima can find all 16
roots of
x^16-12*x^13-4*x^12+54*x^10+36*x^9+6*x^8-108*x^7-108*x^6-36*x^5+75*x^4+108*x^3+54*x^2+18*x+2
(exactly and symbolically), but given that *one* of these roots is big and
messy (see below), it's not clear what the point is....
It can even find roots of some equations like this with symbolic parameters,
but again the result will be huge and hard to interpret in general...
-s
Sample solution:
x =
-sqrt(-((3*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4)*(sqr\
t(-32*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3\
)*sqrt(2*sqrt(43)+6*sqrt(3)))^(3/2)*%i+(96*3^(3/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/2)-1152*(2*sqrt\
(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)+1152*3^(1/4)*(2*sqrt(43)+6*sqrt(3))^(1/3)*sqrt(\
(2*sqrt(43)+6*sqrt(3))^(2/3)-4))*sqrt(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt\
(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))*%i-(192*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1\
/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4)-96*3^(3/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))*(sqrt((2\
*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(\
43)+6*sqrt(3)))-96*3^(1/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(9/4)+576*3^(3/8)*(2*sqrt(43)+6*sqrt(3))^(\
1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(7/4)-1152*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/3)*((2*sqrt(43)+6*s\
qrt(3))^(2/3)-4)^(5/4)+7329*3^(7/8)*sqrt(2*sqrt(43)+6*sqrt(3))*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))\
/(18*3^(7/16)*(2*sqrt(43)+6*sqrt(3))^(1/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/8))+9/2)^(2/3)-2*sqrt(s\
qrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2\
*sqrt(43)+6*sqrt(3)))*%i+2*3^(3/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4)-4*3^(5/8)*(2*sqrt(43)+6*sqrt\
(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4))*sqrt((3*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sq\
rt(43)+6*sqrt(3))^(2/3)-4)^(1/4)*(sqrt(-32*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+\
6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))^(3/2)*%i+(96*3^(3/4)*((2*sqrt(43)+6\
*sqrt(3))^(2/3)-4)^(3/2)-1152*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)+1152*3^(1/\
4)*(2*sqrt(43)+6*sqrt(3))^(1/3)*sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4))*sqrt(sqrt((2*sqrt(43)+6*sqrt(3))\
^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))*%i-(\
192*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4)-96*3^(3/8)*((2*sqrt(43\
)+6*sqrt(3))^(2/3)-4)^(3/4))*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/\
3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))-96*3^(1/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(9/4)\
+576*3^(3/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(7/4)-1152*3^(5/8)*(2*sqrt(\
43)+6*sqrt(3))^(1/3)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(5/4)+7329*3^(7/8)*sqrt(2*sqrt(43)+6*sqrt(3))*((\
2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))/(18*3^(7/16)*(2*sqrt(43)+6*sqrt(3))^(1/4)*((2*sqrt(43)+6*sqrt(3)\
)^(2/3)-4)^(3/8))+9/2)^(2/3)-2*sqrt(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3\
))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))*%i+2*3^(3/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-\
4)^(3/4)-4*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4))/(sqrt(-32*(sqr\
t((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*s\
qrt(43)+6*sqrt(3)))^(3/2)*%i+(96*3^(3/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/2)-1152*(2*sqrt(43)+6*sqr\
t(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)+1152*3^(1/4)*(2*sqrt(43)+6*sqrt(3))^(1/3)*sqrt((2*sqrt(43\
)+6*sqrt(3))^(2/3)-4))*sqrt(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)\
-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))*%i-(192*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sq\
rt(43)+6*sqrt(3))^(2/3)-4)^(1/4)-96*3^(3/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))*(sqrt((2*sqrt(43)+\
6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt\
(3)))-96*3^(1/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(9/4)+576*3^(3/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*s\
qrt(43)+6*sqrt(3))^(2/3)-4)^(7/4)-1152*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/3)*((2*sqrt(43)+6*sqrt(3))^(2\
/3)-4)^(5/4)+7329*3^(7/8)*sqrt(2*sqrt(43)+6*sqrt(3))*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))/(18*3^(7/\
16)*(2*sqrt(43)+6*sqrt(3))^(1/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/8))+9/2)^(1/3))+54*3^(7/16)*(2*sq\
rt(43)+6*sqrt(3))^(1/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/8)*(sqrt(-32*(sqrt((2*sqrt(43)+6*sqrt(3))^\
(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))^(3/2)\
*%i+(96*3^(3/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/2)-1152*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+\
6*sqrt(3))^(2/3)-4)+1152*3^(1/4)*(2*sqrt(43)+6*sqrt(3))^(1/3)*sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4))*sq\
rt(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sq\
rt(2*sqrt(43)+6*sqrt(3)))*%i-(192*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4\
)^(1/4)-96*3^(3/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3\
/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))-96*3^(1/8)*((2*sqrt\
(43)+6*sqrt(3))^(2/3)-4)^(9/4)+576*3^(3/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-\
4)^(7/4)-1152*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/3)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(5/4)+7329*3^(7/8)\
*sqrt(2*sqrt(43)+6*sqrt(3))*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))/(18*3^(7/16)*(2*sqrt(43)+6*sqrt(3)\
)^(1/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/8))+9/2)^(1/3))/((sqrt(-32*(sqrt((2*sqrt(43)+6*sqrt(3))^(2\
/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))^(3/2)*%\
i+(96*3^(3/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/2)-1152*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*\
sqrt(3))^(2/3)-4)+1152*3^(1/4)*(2*sqrt(43)+6*sqrt(3))^(1/3)*sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4))*sqrt\
(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt\
(2*sqrt(43)+6*sqrt(3)))*%i-(192*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^\
(1/4)-96*3^(3/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4\
)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))-96*3^(1/8)*((2*sqrt(4\
3)+6*sqrt(3))^(2/3)-4)^(9/4)+576*3^(3/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)\
^(7/4)-1152*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/3)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(5/4)+7329*3^(7/8)*s\
qrt(2*sqrt(43)+6*sqrt(3))*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))/(18*3^(7/16)*(2*sqrt(43)+6*sqrt(3))^\
(1/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/8))+9/2)^(1/3)*sqrt((3*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*\
((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4)*(sqrt(-32*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqr\
t(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))^(3/2)*%i+(96*3^(3/4)*((2*sqrt\
(43)+6*sqrt(3))^(2/3)-4)^(3/2)-1152*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)+1152\
*3^(1/4)*(2*sqrt(43)+6*sqrt(3))^(1/3)*sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4))*sqrt(sqrt((2*sqrt(43)+6*sq\
rt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3))\
)*%i-(192*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4)-96*3^(3/8)*((2*s\
qrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3\
))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))-96*3^(1/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)\
^(9/4)+576*3^(3/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(7/4)-1152*3^(5/8)*(2\
*sqrt(43)+6*sqrt(3))^(1/3)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(5/4)+7329*3^(7/8)*sqrt(2*sqrt(43)+6*sqrt(\
3))*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))/(18*3^(7/16)*(2*sqrt(43)+6*sqrt(3))^(1/4)*((2*sqrt(43)+6*s\
qrt(3))^(2/3)-4)^(3/8))+9/2)^(2/3)-2*sqrt(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*\
sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))*%i+2*3^(3/8)*((2*sqrt(43)+6*sqrt(3))^\
(2/3)-4)^(3/4)-4*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4))/(sqrt(-3\
2*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sq\
rt(2*sqrt(43)+6*sqrt(3)))^(3/2)*%i+(96*3^(3/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/2)-1152*(2*sqrt(43)\
+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)+1152*3^(1/4)*(2*sqrt(43)+6*sqrt(3))^(1/3)*sqrt((2*s\
qrt(43)+6*sqrt(3))^(2/3)-4))*sqrt(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))\
^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))*%i-(192*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*\
((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4)-96*3^(3/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))*(sqrt((2*sqr\
t(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+\
6*sqrt(3)))-96*3^(1/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(9/4)+576*3^(3/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)\
*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(7/4)-1152*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/3)*((2*sqrt(43)+6*sqrt(\
3))^(2/3)-4)^(5/4)+7329*3^(7/8)*sqrt(2*sqrt(43)+6*sqrt(3))*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))/(18\
*3^(7/16)*(2*sqrt(43)+6*sqrt(3))^(1/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/8))+9/2)^(1/3))))/(2*3^(13/\
16)*(2*sqrt(43)+6*sqrt(3))^(1/12)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/8))-sqrt((3*3^(5/8)*(2*sqrt(43)+\
6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4)*(sqrt(-32*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4\
)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))^(3/2)*%i+(96\
*3^(3/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/2)-1152*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(\
3))^(2/3)-4)+1152*3^(1/4)*(2*sqrt(43)+6*sqrt(3))^(1/3)*sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4))*sqrt(sqrt\
((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sq\
rt(43)+6*sqrt(3)))*%i-(192*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4)\
-96*3^(3/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*\
sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))-96*3^(1/8)*((2*sqrt(43)+6*\
sqrt(3))^(2/3)-4)^(9/4)+576*3^(3/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(7/4\
)-1152*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/3)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(5/4)+7329*3^(7/8)*sqrt(2\
*sqrt(43)+6*sqrt(3))*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/4))/(18*3^(7/16)*(2*sqrt(43)+6*sqrt(3))^(1/4)\
*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/8))+9/2)^(2/3)-2*sqrt(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/\
4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))*%i+2*3^(3/8)*((2*sqr\
t(43)+6*sqrt(3))^(2/3)-4)^(3/4)-4*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4\
)^(1/4))/(sqrt(-32*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/\
4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))^(3/2)*%i+(96*3^(3/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/2)\
-1152*(2*sqrt(43)+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)+1152*3^(1/4)*(2*sqrt(43)+6*sqrt(3)\
)^(1/3)*sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4))*sqrt(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sq\
rt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)*sqrt(2*sqrt(43)+6*sqrt(3)))*%i-(192*3^(5/8)*(2*sqrt(43)+\
6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/4)-96*3^(3/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3\
/4))*(sqrt((2*sqrt(43)+6*sqrt(3))^(2/3)-4)*(3^(3/4)*(2*sqrt(43)+6*sqrt(3))^(2/3)-4*3^(3/4))+12*sqrt(3)\
*sqrt(2*sqrt(43)+6*sqrt(3)))-96*3^(1/8)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(9/4)+576*3^(3/8)*(2*sqrt(43)\
+6*sqrt(3))^(1/6)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(7/4)-1152*3^(5/8)*(2*sqrt(43)+6*sqrt(3))^(1/3)*((2\
*sqrt(43)+6*sqrt(3))^(2/3)-4)^(5/4)+7329*3^(7/8)*sqrt(2*sqrt(43)+6*sqrt(3))*((2*sqrt(43)+6*sqrt(3))^(2\
/3)-4)^(3/4))/(18*3^(7/16)*(2*sqrt(43)+6*sqrt(3))^(1/4)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(3/8))+9/2)^(\
1/3))/(2*3^(13/16)*(2*sqrt(43)+6*sqrt(3))^(1/12)*((2*sqrt(43)+6*sqrt(3))^(2/3)-4)^(1/8))
Numerical value = 1.367537475857394 %i - 0.68209558124294
... but allroots finds this faster