The discussion revolves around finding the value of the expression $$\frac{3}{a}+\frac{3}{a^2}+\frac{3}{a^3}$$ where $$a=\sqrt[3]{4}+\sqrt[3]{2}+1$$. Participants explore different methods of simplifying the expression, considering both substitution and algebraic manipulation.
Participants do not reach a consensus on the best method to simplify the expression. Multiple competing views on how to approach the problem remain, with some favoring direct substitution and others advocating for algebraic manipulation.
Participants express uncertainty regarding the simplification process and the potential for further reduction of the derived expressions. The discussion highlights the complexity of the algebra involved and the different interpretations of the problem.
anemone said:Given $$a=\sqrt[3]{4}+\sqrt[3]{2}+1$$, find the value of $$\frac{3}{a}+\frac{3}{a^2}+\frac{3}{a^3}$$.
Prove It said:Am I missing something, or can we just substitute the value of a? Or are you expecting this to simplify to something simple? :P
Let $\lambda = \sqrt[3]2$, so that $\lambda^3 = 2$. Then $(\lambda-1)a = (\lambda-1)(\lambda^2 + \lambda +1) = \lambda^3-1 = 2-1=1.$ Therefore $1/a = \lambda-1$, and $$\begin{aligned}\frac{3}{a}+\frac{3}{a^2}+\frac{3}{a^3} &= 3\bigl((\lambda-1) + (\lambda-1)^2 + (\lambda-1)^3\bigr) \\ &= 3(\lambda-1)(\lambda^2 - \lambda + 1) \\ &= 3(\lambda^3 -2\lambda^2 +2\lambda - 1) \\ &= 3(1+2\lambda-2\lambda^2) \\ &= 3 + 6\sqrt[3]2 - 6\sqrt[3]4.\end{aligned}$$ I don't see that it can be simplified further than that.anemone said:Given $$a=\sqrt[3]{4}+\sqrt[3]{2}+1$$, find the value of $$\frac{3}{a}+\frac{3}{a^2}+\frac{3}{a^3}$$.