Solving Complex Equations Involving Roots of Unity

When dealing with complex equations, it is often necessary to employ advanced algebraic techniques, including the use of roots of unity, to find solutions. This article will walk you through the process of solving a specific equation that combines polynomial terms, fractional exponents, and roots of unity.

Introduction

Consider the equation given by:

[ x^3 frac{1}{x^3} sqrt{x} frac{1}{sqrt{x}} ]

Note that this simplifies to:

[ x^6 - 1 x - frac{1}{x} ]

Transformation and Simplification

To solve the equation more easily, we first make a substitution. Let ( sqrt{x} u ). This substitution transforms our original equation into:

[ u^6 - frac{1}{u^6} u - frac{1}{u} ]

Multiplying through by ( u^6 ) to clear the denominators gives:

[ u^{12} - u^7 - u^5 1 0 ]

This polynomial equation can be factored as:

[ (u^7 - 1)(u^5 - 1) 0 ]

Setting each factor to zero yields:

[ u^7 1 ] and [ u^5 1 ]

The solutions to these equations are the 7th and 5th roots of unity, respectively. Specifically, we have:

[ u e^{frac{2pi in}{7}} ] for ( n 0, 1, 2, 3, 4, 5, 6 )

[ u e^{frac{2pi ik}{5}} ] for ( k 0, 1, 2, 3, 4 )

Since ( u sqrt{x} ), the solutions for ( x ) are:

[ x e^{frac{4pi in}{7}} ] for ( n 0, 1, 2, 3, 4, 5, 6 )

[ x e^{frac{4pi ik}{5}} ] for ( k 0, 1, 2, 3, 4 )

These solutions correspond to the even powers of the 5th and 7th roots of unity.

Verification and Additional Insights

To verify that these are the complete set of solutions, we can consider the behavior of the function ( f(x) x^6 - x^{-6} - x - x^{-1} ) for positive ( x ). Taking the derivative:

[ f'(x) 6x^5 6x^{-7} - 1 - x^{-1} ]

Setting the derivative equal to zero and solving for ( x ) gives:

[ 6x^{12} - x^6 - x^5 - 6 0 ]

The only positive solution to this equation is ( x 1 ), as confirmed by the fact that the remaining terms are always positive for positive ( x ).

Therefore, ( x 1 ) is a solution, and we have covered all possible solutions by considering the roots of unity.

Conclusion

By using the concept of roots of unity and algebraic techniques, we can effectively solve complex equations involving fractional exponents and polynomial terms. This method not only provides a rigorous framework for solving such equations but also deepens our understanding of the underlying mathematical structures.