There are precisely two nonnegative real values of R, so that for each of these values:

³√(3 + √R) + ³√(3 - √R) is an integer.

Find these values, and prove that no other nonnegative real R can conform to the given conditions.

*** While the solution may be facile with the aid of a computer program, show how to derive it without one.

I don't know how to script the nifty root and exponentiation characters, so this is mostly text.

If you let A=first cube root and B=second cube root and I=integer, then cube the equation, you find the 2 cubes sum to 6 and (A+B)=I factors remaining expression, so I factors 6.

For I=1, R = 368/27.

For I=2, R= 242/27.

For I=3, R= -100/27.

For I=6, R= -42652/27.

Only the first 2 solution satisfy the problem's conditions.

That's 4 solutions in all, which seems weird, so maybe I've made a mistake.

Posted by xdog on 2008-08-08 21:12:39