MHB Find the value of 3/a + 3/a² + 3/a³.

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To find the value of 3/a + 3/a² + 3/a³ with a = √[3]{4} + √[3]{2} + 1, participants discuss substitution versus simplification methods. One approach involves letting λ = √[3]{2} and deriving a formula for a, leading to a more complex expression. Another participant notes that a can be interpreted as the sum of a geometric series, allowing for a different substitution that simplifies the expression. Ultimately, both methods yield similar results, indicating that the expression cannot be simplified further. The discussion highlights the importance of exploring multiple approaches to solve the problem effectively.
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Given $$a=\sqrt[3]{4}+\sqrt[3]{2}+1$$, find the value of $$\frac{3}{a}+\frac{3}{a^2}+\frac{3}{a^3}$$.
 
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Re: Find the value of 3/a+3/a²+3/a³.

anemone said:
Given $$a=\sqrt[3]{4}+\sqrt[3]{2}+1$$, find the value of $$\frac{3}{a}+\frac{3}{a^2}+\frac{3}{a^3}$$.

Am I missing something, or can we just substitute the value of a? Or are you expecting this to simplify to something simple? :P
 
Re: Find the value of 3/a+3/a²+3/a³.

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

Hi Prove It, I really think we have to substitute the value of a into the intended expression and further simplifying it from there...:)
 
Re: Find the value of 3/a+3/a²+3/a³.

anemone said:
Given $$a=\sqrt[3]{4}+\sqrt[3]{2}+1$$, find the value of $$\frac{3}{a}+\frac{3}{a^2}+\frac{3}{a^3}$$.
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.
 
Re: Find the value of 3/a+3/a²+3/a³.

Thanks Opalg for participating in this problem and my answer is 'quite' similar to yours too.:)

My solution:

I noticed that $$a=\sqrt[3]{4}+\sqrt[3]{2}+1=1+\sqrt[3]{2}+\sqrt[3]{4}$$ is actually the sum of the first three terms of a geometric progression with first term and common ratio to be 1 and $$\sqrt[3]{2}$$ respectively and I found another way to rewrite $$a$$, i.e.

$$a=S_3=\frac{1((\sqrt[3]{2})^3-1)}{\sqrt[3]{2}-1}=\frac{1}{\sqrt[3]{2}-1}$$.

Therefore, the intended expression could be found by substituting this formula for $$a$$ into it to get:

$$\frac{3}{a}+\frac{3}{a^2}+\frac{3}{a^3}=\frac{3}{a^3}\left(1+a+a^2\right)=3(\sqrt[3]{2}-1)((\sqrt[3]{2})^2-\sqrt[3]{2}+1)$$

My little note to Prove It:

I am truly sorry, Prove It for telling you to substitute the value of $$a$$ straight into the equation and then to do the simplification, because what I am posting now is totally going against what I said to you but on the level, I did try to approach it using two ways and the initial one was to substitute first and simplify next.
 
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