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

  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    Algebra Challenge
AI Thread Summary
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.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
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 $b$.
 
Mathematics news on Phys.org
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...
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

Similar threads

Back
Top