8

I'm wondering if there is a way to simplify the nested cubic radicals

$$\sqrt[3]{\sqrt[3]{A}-B}$$ into its denested form $$\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$$

Examples include $$\sqrt[3]{7\sqrt[3]{20}-1}=\sqrt[3]{\frac {16}{9}}+\sqrt[3]{\frac {100}{9}}-\sqrt[3]{\frac {5}{9}}$$ $$\sqrt[6]{7\sqrt[3]{20}-19}=\sqrt[3]{\frac {5}{3}}-\sqrt[3]{\frac {2}{3}}$$ which seem to have the denested form. Is there a systematic procedure?

Quanto
  • 97,352
Frank
  • 5,984
  • 2
    I don't think it's possible for every nested radical of this form, but in some cases sure. Maybe just expand $$(\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c})^3$$ and try to set the conditions to get this into a form $$\sqrt[3]{A}-B$$ – Yuriy S May 28 '16 at 20:02
  • So far, it seems like $\sqrt[3]{7\sqrt[3]{20}-B}$ can be simplified into the form $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$... – Frank May 28 '16 at 20:30
  • I'm sure it's possible for other values of $A$ as well. Just not every one – Yuriy S May 28 '16 at 20:32
  • 1
    Related http://math.stackexchange.com/questions/871639/denesting-radicals-like-sqrt3-sqrt32-1?rq=1 – cgiovanardi May 28 '16 at 22:09
  • 2
    Another one http://math.stackexchange.com/questions/194030/simplify-sqrt-sqrt35-sqrt34/194148#194148 – cgiovanardi May 28 '16 at 22:25

1 Answers1

7

The answer is yes, curtesy of Ramanujan.

For nested cubic radicals of the form $\sqrt[3]{\sqrt[3]{A}-B}$, denote $C =\sqrt[3]{B^3-A}$ and construct the corresponding cubic polynomial below $$R(x)=x^3 + \frac{B+2C}3x^2 - \frac{(B-C)(2B+C)}{27}x+ \frac{(B-C)^3}{729}$$ which, per Ramanujan, has the property that its roots $x_1$, $x_2$ and $x_3$ satisfy $$\sqrt[3]{x_1}+ \sqrt[3]{x_2 }+ \sqrt[3]{x_3 } =\sqrt[3]{\sqrt[3]{A}-B} $$ Thus, denesting $\sqrt[3]{\sqrt[3]{A}-B}$ reduces to solving the cubic equation $R(x)=0$, a straightforward exercise. Take the example of $\sqrt[3]{21\sqrt[3]{6}-17}$, with $B=17$ and $C= -37$. Its denesting polynomial is $$x^3 -19x^2+6x + 216 =(x-18)(x-4)(x+3)$$ which results in $$\sqrt[3]{21\sqrt[3]{6}-17}=\sqrt[3]{18}+\sqrt[3]{4}-\sqrt[3]{3}$$

Listed below are other examples admitting rational detestation, along with their respective polynomials: $$ \begin{align} \sqrt[3]{\sqrt[3]{2}-1}&=\sqrt[3]{\frac19}-\sqrt[3]{\frac29}+\sqrt[3]{\frac49}, &\>\>\> &x^3 -\frac13 x^2-\frac2{27}x + \frac8{729}\\ \sqrt[3]{39\sqrt[3]{12}-19}&=\sqrt[3]{48}+\sqrt[3]{9}-\sqrt[3]{4}, &&x^3 -53x^2+204x + 1728\\ \sqrt[3]{7\sqrt[3]{20}-1} &=\sqrt[3]{\frac {100}9}+\sqrt[3]{\frac {16}9}-\sqrt[3]{\frac 59}, && x^3 -\frac{37}3x^2+\frac{340}{27}x + \frac{8000}{729}\\ \sqrt[3]{93\sqrt[3]{30}+19}&=\sqrt[3]{180}+\sqrt[3]{25}-\sqrt[3]{6}, &&x^3 -199 x^2+3270 x + 27000 \\ \sqrt[3]{13\sqrt[3]{70}-17}&=\sqrt[3]{\frac{245}9}+\sqrt[3]{\frac{50}9} -\sqrt[3]{\frac{28}9}, &&x^3 -\frac{89}3 x^2+\frac{1330}{27} x + \frac{343000}{729}\\ \end{align} $$

Quanto
  • 97,352