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

[anemone]

Given that \(\displaystyle a,\,b\in\Bbb{N}\) such that \(\displaystyle \sqrt{a^2+2b+1}+\sqrt[3]{b^3+3a^2+3a+1}\) is a rational number.

Find the relationship between $a$ and $b$.

 

[soroban]

given that \(\displaystyle a,\,b\in N\) such that \(\displaystyle \sqrt{a^2+2b+1}+\sqrt[3]{b^3+3a^2+3a+1}\) 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]

 

[anemone]

[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

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square numbers and inequalities...