Algebra problem that I've been trying to solve
 

Does anyone have an equation for the value of x in this equation?

x^5+16*x^6+243*x^7=1

I have an approximation for the real root, but I want an exact formula for it, even if it includes an infinite series:

[code]0.446223522318958869779779205880610165343221619229686782382108709788[/code]

[QUOTE=Stargate38;374632]Does anyone have an equation for the value of x in this equation?

x^5+16*x^6+243*x^7=1[/QUOTE]
Wolfram Alpha?
[url]http://www.wolframalpha.com/input/?i=x%5E5%2B16*x%5E6%2B243*x%5E7%3D1[/url]

Offhand I would guess !∃ a closed-form expression for the single real root, but perhaps it will be of use to note that the expression can be rewritten as

x^4*(x + 2^4*x^2 + 3^5*x^3) = 1 , or alternatively

x^2*(1^3*x^3 + 2^4*x^4 + 3^5*x^5) = 1 .

(This is unhelpful in terms of directly leading to a closed-form for the real root, but perhaps this kind of equation has been previously studied in some particular mathematical context.)

[QUOTE=Stargate38;374632]Does anyone have an equation for the value of x in this equation?

x^5+16*x^6+243*x^7=1

I have an approximation for the real root, but I want an exact formula for it, even if it includes an infinite series:

[code]0.446223522318958869779779205880610165343221619229686782382108709788[/code][/QUOTE]

best I can think of is:

x^5+16*x^6+243*x^7 = 1;
x^5+16*(x^5)^1.2+243*(x^5)^1.4 = 1;
1+16*(x^5)^0.2 + 243*(x^5)^0.4 = 1/(x^5);
16*(x^5)^0.2+243*(x^5)^0.4 = (1-x^5)/(x^5);

[QUOTE=ewmayer;374649]Offhand I would guess !∃ a closed-form expression for the single real root, but perhaps it will be of use to note that the expression can be rewritten as

x^4*(x + 2^4*x^2 + 3^5*x^3) = 1 , or alternatively

x^2*(1^3*x^3 + 2^4*x^4 + 3^5*x^5) = 1 .

(This is unhelpful in terms of directly leading to a closed-form for the real root, but perhaps this kind of equation has been previously studied in some particular mathematical context.)[/QUOTE]

Sigh. People herein still refuse to learn math or use Google.
It is nauseating. How hard can it be to look up "septic equation"?????

Hint: It is an irreducible Septic.

Yes, there is a closed form solution. But it is not algebraic and
it is not elementary. It can be solved in terms of Theta functions.

[QUOTE=R.D. Silverman;374714]Sigh. People herein still refuse to learn math or use Google.
It is nauseating.

Hint: It is an irreducible Septic.

Yes, there is a closed form solution. But it is not algebraic and
it is not elementary. It can be solved in terms of Theta functions.[/QUOTE]

Oh. It is very bad form to present a polynomial this way.

[QUOTE=R.D. Silverman;374714]Sigh. People herein still refuse to learn math or use Google.
It is nauseating. How hard can it be to look up "septic equation"?????[/QUOTE]

I did, but got a raft of links to antibiotic creams and iodine compounds. My google search history must differ from yours, for the results we see to be so wildly different.

[QUOTE=R.D. Silverman;374714]Hint: It is an irreducible Septic.

Yes, there is a closed form solution. But it is not algebraic and
it is not elementary. It can be solved in terms of Theta functions.[/QUOTE]

Not all irreducible septics are unsolvable, of course, but you're right that this one is. Its Galois group is [TEX]S_7.[/TEX]

[QUOTE=ewmayer;374725]I did, but got a raft of links to antibiotic creams and iodine compounds. My google search history must differ from yours, for the results we see to be so wildly different.[/QUOTE]

:snicker:

[QUOTE=CRGreathouse;374727]Not all irreducible septics are unsolvable, of course, but you're right that this one is. Its Galois group is [TEX]S_7.[/TEX][/QUOTE]

C7 would have been nicer........

[QUOTE=TheMawn;374733]:snicker:[/QUOTE]

Uh, huh huh huh, uh [URL=http://www.wikipedia.org/wiki/Beavis and Butthead]Beavis[/URL], you said "septic", uh huh huh huh...

Whoa! They're like talking about poop in math class! Yeah! Yeah! Galois theory *kicks ass*!