Eliminating Multiple Radicals in an Equation
Date: 02/23/2006 at 16:26:22 From: Jim Subject: eliminating cube roots and higher roots in equations. Can you simplify an expression that adds cube roots or higher roots, i.e. eliminate the radicals from the equation? For example: Cuberoot(x) + Cuberoot(y) = 1 or Cuberoot(x) + Fourthroot(y) = 1 It is clear how to proceed with square roots: move the nonradical terms to one side of the equation, square, and repeat; however, with cube roots, we end up with cross terms that seem impossible to get rid of.
Date: 02/23/2006 at 17:29:01 From: Doctor Vogler Subject: Re: eliminating cube roots and higher roots in equations. Hi Jim, Thanks for writing to Dr. Math. That's not an easy task, but here is one way to solve the problem: First solve for cuberoot(y) = 1 - cuberoot(x) or fourthroot(y) = 1 - cuberoot(x) and then raise to the 3rd or 4th power to get rid of the root on the left. On the right side, if you change cuberoot(x)^3 into x, then you should have a quadratic polynomial in cuberoot(x). Solve for it using the quadratic formula. This will introduce a square root that we didn't have before, so we'll deal with that later. Now raise both sides to the 3rd power to get rid of the cube root, and then solve for the square root that arose from the quadratic formula. Square both sides, and you are done! If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/
Date: 02/23/2006 at 21:11:53 From: Jim Subject: eliminating cube roots and higher roots in equations. Dr. Vogler: Thanks for your answer. Can you tell me if you can extend this idea to additional terms, something like: cuberoot(x) + fifthroot(y) + seventhroot(z) = 0 I thought you should solve for the highest power, seventhroot(z) = fifthroot(y) - cuberoot(x) then raise both sides to 7th power, but I end up with cross terms that don't seem to be able to simplified. Then I thought to find the lowest common multiple (105) and raise to that power, so for at least one term, the radicals disappear. But as you can see the problem just blows up into more cross terms. Is it impossible? I'm just looking for the kernel of the idea of how this problem would be approached. Thanks, Jim
Date: 02/24/2006 at 16:40:33 From: Doctor Vogler Subject: Re: eliminating cube roots and higher roots in equations. Hi Jim, Here is a completely different way to solve your problem that works better in general. Methods like the one I described last time are pretty ad-hoc; i.e. they don't generalize well. Instead, you can use linear algebra, and this idea will work for polynomial equations in lots of different roots. Consider your roots as a, b, c. That is, a = cuberoot(x) b = fifthroot(y) c = seventhroot(z) which we would do better to write as a^3 = x b^5 = y c^7 = z (better, because these are polynomial equations). Then you want to convert the equation a + b + c = 0 into a polynomial equation in x, y, and z alone. So the thing to do is to multiply this equation by every monomial a^i * b^j * c^k with 0 <= i < 3 0 <= j < 5 0 <= k < 7 and replace every occurrence of a^3 by x, of b^5 by y, of c^7 by z. For example, the first few equations would be a^2 + ab + ac = 0 x + a^2*b + a^2*c = 0 ab + b^2 + bc = 0 a^2*b + ab^2 + abc = 0 ... Now if you treat x, y, and z as constants and consider only the variables a, b, and c, then you will end up with 3*5*7 polynomial equations in the 3*5*7 monomials a^i * b^j * c^k. In other words, if you consider each monomial as a variable, then you can write these in matrix form as A*v = 0 where A is a square matrix of height and width 3*5*7 which contains the variables x, y, and z but none of a, b, and c, and v is a column vector that lists off the 3*5*7 monomials in a, b, and c. Unless the only solution to your equations is a = b = c = 0, that means that the matrix A is not invertible, and therefore det(A) = 0. But the determinant is a polynomial function in the entries of your matrix, which means that the determinant of A is a polynomial in x, y, and z. Conclusion: If you form the matrix by multiplying your equation by all of the monomials in the roots, then the determinant of that matrix gives you a polynomial equation in your variables x, y, and z. Of course, taking the determinant of a 105-by-105 matrix is no easy task, so I would recommend having a computer program do the work for you. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/
Date: 02/26/2006 at 13:48:12 From: Jim Subject: Thank you (eliminating cube roots and higher roots in equations.) Thank you, Dr. Vogler. This is a very interesting approach to the problem that I don't think I would have found anywhere else. - Jim
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