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Find positive integers for both a and b

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,687
I have a question relating to solving for both a and b in the following question:

Find positive integers a and b such that:

$\displaystyle \left(\sqrt[3]{a}+\sqrt[3]{b}-1 \right)^2=49+20\sqrt[3]{6}$

This one appears to be tough because it doesn't seem right to expand the left hand side and I have tried that but it lead me to nowhere closer to finding both integers values for a and b.

Also, I've tried to work on the right hand side as my effort revolved around rewriting it as the square of sum of two terms but this has not been a fruitful approach as well.

Do you guys have any idea on how I can solve this one, please?

Thanks in advance.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Find positive integers a and b such that:

$\displaystyle \left(\sqrt[3]{a}+\sqrt[3]{b}-1 \right)^2=49+20\sqrt[3]{6}$
One way: expressing $\sqrt{49+20\sqrt[3]{6}}=\alpha +\beta \sqrt[3]{6}+\gamma \sqrt[3]{36}$ with $\alpha,\beta,\gamma\in\mathbb{Q}$ we get $\alpha=-1,\beta=\gamma=2$. Now, identifyng, we obtain $a=48,b=288$ (0r reciprocally).

P.S. For the first step, we have used that a basis of $\mathbb{Q}(\sqrt[3]{6})$ over $\mathbb{Q}$ is $B=\{1,\sqrt[3]{6},\sqrt[3]{36}\}$. I ignore if at Olympiad level the students cover some similar property (of course without using the theory of extension fields).