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Algebra Level 5

Let $\Lambda$ be a real number that can be represented in the nested radical form as

$\Lambda = \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \cdots}}}$

If the closed form of $\Lambda$ is

$\Lambda = \displaystyle {\frac {{\sqrt[{3}]{a+b{\sqrt {c}}}}+{\sqrt[{3}]{a-b{\sqrt {c}}}}}{d}}$

where $a, b, c$ and $d$ are positive integers with $c$ square-free. Find the smallest possible value of $a+b+c+d$ .

1 solution

Ραμών Αδάλια
Mar 26, 2017

The answer is given by the real root of the polynomial $x^3-x-1$ , which is $\frac{\sqrt[3]{108+12\sqrt{69}}+ \sqrt[3]{108-12\sqrt{69}} }{6}.$

How did you get $\dfrac{\sqrt[3]{108+12\sqrt{69}}+ \sqrt[3]{108-12\sqrt{69}} }{6}$ in the first place?

Pi Han Goh - 4 years, 2 months ago

I found the roots of the depressed cubic polynomial using the standard method of substituing the variable by something else, guess what ;)

Ραμών Αδάλια - 4 years, 2 months ago

If so, then it would be better to state that in your solution, so that it's easier for others to understand how you obtained the final result.

Note that what you just did is called Cardano's method .

Pi Han Goh - 4 years, 2 months ago