MHB Can You Find the Relationship Between a and b in This Algebra Problem?

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The discussion centers on finding the relationship between two natural numbers, a and b, based on the expression involving square and cube roots. It is noted that if a equals b, both radicals simplify to a + 1, which is a rational number. Participants agree that this observation is valid and point out that further proof is needed to establish this relationship formally. The conversation emphasizes the importance of logical reasoning in proving the equality of the radicals. Ultimately, the relationship hinges on the condition that a must equal b for the expression to remain rational.
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Given that $$a,\,b\in\Bbb{N}$$ such that $$\sqrt{a^2+2b+1}+\sqrt[3]{b^3+3a^2+3a+1}$$ is a rational number.

Find the relationship between $a$ and $b$.
 
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anemone said:
given that [math]a,\,b\in N[/math] such that [math]\sqrt{a^2+2b+1}+\sqrt[3]{b^3+3a^2+3a+1}[/math] is a rational number.

Find the relationship between $a$ and $b$.
[sp]
By inspection, we see that, if a=b, the two radicals are equal to a+1.

[/sp]

 
soroban said:
[sp]
By inspection, we see that, if a=b, the two radicals are equal to a+1.

[/sp]

That's correct, soroban!(Cool)

All that's left now is its proof which can be done by applying some thought on the
square numbers and inequalities...
 
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