The discussion focuses on solving the equation ##x^{1/6}=12x^{1/3}-1## and explores various algebraic techniques and substitutions, particularly using ##u=x^{1/6}##. Participants confirm that the equation can be transformed into a cubic form, leading to solutions such as ##u=2##, which corresponds to ##x=64##. The conversation emphasizes the importance of verifying solutions against the original equation and discusses the limitations of finding closed-form solutions for polynomials of degree greater than four, suggesting numerical methods for such cases.
PREREQUISITESMathematicians, algebra students, and anyone interested in solving complex equations involving surds and polynomial expressions.
sorry for not seeing then, ...##u=2##fresh_42 said:Yes, and ##12=12\cdot 1 = 6 \cdot 2 = 4 \cdot 3##. Which one do we get for ##u##?