[superprismatic]

cuberoot(3298 + 627550 * sqrt(-5)) +

cuberoot(3298 - 627550 * sqrt(-5))

represents a certain positive integer

N. If the digits of N are converted

to letters (A=1, B=2, etc.), They

spell a certain word. What is it?

[Guest]

Wonder how I could "aid" in solving this problem! Am I right?

Edited September 3, 2009 by DeeGee

[superprismatic]

Wonder how I could "aid" in solving this problem! Am I right?

Right as rain!

[Guest]

can you hint how you got rid of i?

I can't seem to do it ;-)

[bushindo]

can you hint how you got rid of i?

I can't seem to do it ;-)

Here's the hint

How to take roots of complex numbers-

De Moivre's formula

Edited September 3, 2009 by bushindo

[Guest]

But when i got down to it, it ended up being:

2*cuberoot(3298 +/- i*627550*sqrt(5))

that gives r = 12534*sqrt(12534)

and a theta of 89.865 yada yada.

which means that the cube root is 97 +/- i*55.9016 or so. Mult by 2, and you get 194 +/- i*111.80 or so.

I got as far as that, and that's where my question came in...how did/do you drop the imaginary part so that you could represent all that above as a positive integer???

[bushindo]

But when i got down to it, it ended up being:

2*cuberoot(3298 +/- i*627550*sqrt(5))

that gives r = 12534*sqrt(12534)

and a theta of 89.865 yada yada.

which means that the cube root is 97 +/- i*55.9016 or so. Mult by 2, and you get 194 +/- i*111.80 or so.

I got as far as that, and that's where my question came in...how did/do you drop the imaginary part so that you could represent all that above as a positive integer???

I think the bolded part is the problem here

I assume you meant that

cuberoot(3298 + 627550 * sqrt(-5)) + cuberoot(3298 - 627550 * sqrt(-5)) = 2 * cuberoot(3298 +/- i*627550*sqrt(5))

The above isn't true because you can not distribute outside the cuberoot sign. The straight forward approach would be to compute all 3 roots of cuberoot(3298 + 627550 * sqrt(-5)), then compute all 3 roots of cuberoot(3298 - 627550 * sqrt(-5)). Looking at those, you'll probably find a pair such that their sum is a positive integer with no imaginary part. Computing the 3 cube roots is easier if you draw it on a graph.

Edited September 3, 2009 by bushindo

[superprismatic]

But when i got down to it, it ended up being:

2*cuberoot(3298 +/- i*627550*sqrt(5))

that gives r = 12534*sqrt(12534)

and a theta of 89.865 yada yada.

which means that the cube root is 97 +/- i*55.9016 or so. Mult by 2, and you get 194 +/- i*111.80 or so.

I got as far as that, and that's where my question came in...how did/do you drop the imaginary part so that you could represent all that above as a positive integer???

I'm not sure what 2*cuberoot(3298 +/- i*627550*sqrt(5)) means. But the original problem has it as

cuberoot(3298 + i*627550*sqrt(5)) + cuberoot(3298 - i*627550*sqrt(5)). When I did the problem I started by cubing the monster then staring at the result and saw a way to simplify (without De Moiuvre's formula, by the way). I ended up with a cubic equation which I could solve.

[Guest]

i get (97 + 56i + 97 - 56i) = 194 --> AID

[Guest]

I think the bolded part is the problem here

I assume you meant that

cuberoot(3298 + 627550 * sqrt(-5)) + cuberoot(3298 - 627550 * sqrt(-5)) = 2 * cuberoot(3298 +/- i*627550*sqrt(5))

The above isn't true because you can not distribute outside the cuberoot sign. The straight forward approach would be to compute all 3 roots of cuberoot(3298 + 627550 * sqrt(-5)), then compute all 3 roots of cuberoot(3298 - 627550 * sqrt(-5)). Looking at those, you'll probably find a pair such that their sum is a positive integer with no imaginary part. Computing the 3 cube roots is easier if you draw it on a graph.

The equation has 4 answers, 2 of which are the same...

cuberoot(3298 + 627550 * i * sqrt(5)) + cuberoot(3298 + 627550 * i * sqrt(5))

cuberoot(3298 + 627550 * i * sqrt(5)) + cuberoot(3298 - 627550 * i * sqrt(5))

cuberoot(3298 - 627550 * i * sqrt(5)) + cuberoot(3298 + 627550 * i * sqrt(5))

cuberoot(3298 - 627550 * i * sqrt(5)) + cuberoot(3298 - 627550 * i * sqrt(5))

The key answers are the middle two...that's where you wind up with principal roots of 97 + 55.901i + 97 - 55.901i

= 194

[superprismatic]

I cubed N, then found that N³ = 37602 * N + 6596. The number I'm after, then, must be a root of this equation (I may have introduced extraneous roots, though). Solving this, I get 194, -193.824..., and -0.175... I am now done. Voila!