Why is $\sqrt[3]{18+5\sqrt{13}} + \sqrt[3]{18-5\sqrt{13}} = 3$?

Question

How to show that $$\sqrt[3]{18+5\sqrt{13}} + \sqrt[3]{18-5\sqrt{13}} = 3?$$

This equality comes from solving $$t^3 - 15 t - 4 = 0$$ using Cardanos fomula and knowing the solution $t_1=4$.

I have attempted multiplying the whole thing with $(\sqrt[3]{18+5\sqrt{13}})^2 - (\sqrt[3]{18-5\sqrt{13}})^2$, but no success. Then I have solved for one cubic root and put all to the third power. Also no success.

The "trick" $(a+b)(a-b) = a^2-b^2$ that works for square roots will not work for cube roots. Maybe you want to think about the analogous formula $(a+b)(a^2-ab+b^2) = a^3+b^3$. — Ted Shifrin, Sep 23 '17 at 16:45
The other two solutions to the equation are $-2\pm\sqrt{3}$, so this is perhaps some clever use of Vieta's formulas. — mechanodroid, Sep 23 '17 at 16:51
The more recent question How to prove that $(-18+\sqrt{325})^{\frac{1}{3}}+(-18-\sqrt{325})^{\frac{1}{3}} = 3$ is very, very nearly a duplicate of this one. — Calum Gilhooley, Mar 07 '19 at 01:02

score 5 · Answer 1 · answered Sep 23 '17 at 16:46

5

Hint :$$a=\sqrt[3]{18+5\sqrt{13}} + \sqrt[3]{18-5\sqrt{13}}=b+c \\a^3=b^3+c^3+3bc(b+c)\\ a^3=18+5\sqrt{13}+18-5\sqrt{13}+3\sqrt[3]{18+5\sqrt{13}}.\sqrt[3]{18-5\sqrt{13}}(a)\\a^3=36+3\sqrt[3]{324-325}a\\a^3=36-3a$$solve for a $$a^3+3a-36=(a-3)(a^2+3a+12)=0 \to a=3$$

answered Sep 23 '17 at 16:46

Khosrotash

24,922

If you remove the dot (its less than 6 char edit) then its the answer. – Raphael J.F. Berger Sep 23 '17 at 16:55

user8277998 · Accepted Answer · 2017-09-23T17:03:50.937

5

Let $(a + b\sqrt{13})^3 = (18 + 5\sqrt{13})$ for $a, b \in \Bbb Q$

Expanding the LHS gives,

$$(a^3 + 39 ab^2 - 18 ) +\sqrt{13}(3a^2 b + 13 b^3 - 5) = 0$$,

From this we get,

$$\begin{cases}a^3 + 39 ab^2 - 18 = 0 \\ 3a^2 b + 13 b^3 - 5 = 0\end{cases}$$

Solving the system give $ a = \dfrac 32$ and $ b = \dfrac12$

Therefore

$$\sqrt[3]{(18 + 5\sqrt{13})} = \dfrac 32 +\dfrac12\sqrt{13}$$

Similarly,

$$\sqrt[3]{(18 - 5\sqrt{13})} = \dfrac 32 -\dfrac12\sqrt{13}$$

Hence the sum is $3$.

edited Sep 23 '17 at 17:03

answered Sep 23 '17 at 16:55

user8277998

2,666

Thats very nice! – Raphael J.F. Berger Sep 23 '17 at 16:59
@R_Berger My reaction when I first learnt this trick. – user8277998 Sep 23 '17 at 17:01
That gives some idea on field extensions, and contains in deed potential to generalization, I think. – Raphael J.F. Berger Sep 23 '17 at 17:48
@MichaelRozenberg I don't doubt that it is a bit ugly . I outlined the general method which OP can use elsewhere, for instance when he does not know the sum beforehand or when he had to denest radicals. No doubt your method is a bit easy here and I had +1 on your answer anyways. – user8277998 Sep 23 '17 at 17:56
@MichaelRozenberg No problem. Cheers :). – user8277998 Sep 23 '17 at 18:10
May I know how did you solve the system? By solving a cubic equation or some other way? – Nov 11 '17 at 19:22
@user486600 I used rational root theorem. – user8277998 Nov 11 '17 at 19:54
That is, you've arrived to a cubic equation and then have used the rational root theorem? – Nov 11 '17 at 19:59
@user486600 Yes, I checked 3 or 4 numbers by hand and got $b^3 = 1/8$, I was lucky I guess. – user8277998 Nov 11 '17 at 20:00
@123 Thanks.... – Nov 11 '17 at 20:02
@user486600 Welcome :). – user8277998 Nov 11 '17 at 20:22

Michael Rozenberg · Answer 3 · 2017-09-23T16:58:25.413

$$\left(\sqrt[3]{18+5\sqrt{13}} + \sqrt[3]{18-5\sqrt{13}}\right)^3=$$ $$=18+5\sqrt{13}+18-5\sqrt{13}+3\sqrt[3]{18+5\sqrt{13}} \sqrt[3]{18-5\sqrt{13}}\left(\sqrt[3]{18+5\sqrt{13}} + \sqrt[3]{18-5\sqrt{13}}\right)=$$ $$=36+3\sqrt[3]{-1}\left(\sqrt[3]{18+5\sqrt{13}} + \sqrt[3]{18-5\sqrt{13}}\right)=$$ $$=36-3\left(\sqrt[3]{18+5\sqrt{13}} + \sqrt[3]{18-5\sqrt{13}}\right).$$ Now, let $\sqrt[3]{18+5\sqrt{13}} + \sqrt[3]{18-5\sqrt{13}}=x.$

Thus, $$x^3=36-3x$$ or $$x^3-3x^2+3x^2-9x+12x-36=0$$ or $$(x-3)(x^2+3x+12)=0,$$ which gives $x=3$.

But I think the best way in this formulation it's the way by using $$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc).$$ Now, we need to prove that $$18+5\sqrt{13}+18-5\sqrt{13}-27-3\cdot(-3)\sqrt[3]{18+5\sqrt{13}}\cdot\sqrt[3]{18-5\sqrt{13}}=0$$ or $$36-27-3(-3)(-1)=0,$$ which is true.

Done!

Thanks guys! I fixed. – Michael Rozenberg Sep 23 '17 at 17:01 — Michael Rozenberg, Sep 23 '17 at 17:01

score 2 · Answer 4 · edited Sep 23 '17 at 16:59

2

Because the function $\mathbb{R} \to \mathbb{R}: x \mapsto x^3$ is injective, it follows that

$x = y \iff x^3 = y^3$ (more precicely: $\Leftarrow$ follows from injectivity)

So, you can cube both sides without any danger, calculate, and make the conclusion you are looking for.

edited Sep 23 '17 at 16:59

Chappers

67,606

answered Sep 23 '17 at 16:46

2

In the square case, I would have taken the danger on me. – Raphael J.F. Berger Sep 23 '17 at 16:50
Seems like we have a real badass here :P – Sep 23 '17 at 20:18

Why is $\sqrt[3]{18+5\sqrt{13}} + \sqrt[3]{18-5\sqrt{13}} = 3$?

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