Find the value of this expression

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The expression to evaluate is x^3 - 6x^2 + 6x, where x = 2 + 2^(2/3) + 2^(1/3). Substituting x directly into the expression proves complicated, prompting suggestions to simplify calculations by breaking down x^2 and x^3 first. A more efficient approach involves rewriting the expression as (x-2)^3 - 6x + 8, which significantly reduces the complexity of the calculations. This method allows for easier simplification and ultimately saves time in solving the problem. The discussion emphasizes the importance of strategic simplification in algebraic expressions.
utkarshakash
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Homework Statement


If x=2+2^{2/3}+2^{1/3}, then the value of x^3-6x^2+6x is

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The Attempt at a Solution


The very first idea that comes to my mind is to substitute the value of x in the given expression. But that gets very complicated and long as well. Any other ideas?
 
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I don't think you can avoid this. Simplify as early and as much as possible (e.g. calculate x^2, simplify that and multiply with x to get x^3).
 
It doesn't look so bad...
x3 is easy since 3s cancel a lot. Then x2 isn't too bad because 2*2 +2*1 = 6 which divides by 3 easily.

It all simplifies nicely as you go along.:smile:
 
You can simplify the calculations a little by writting this as x(6+ x(x- 6)).
 
Try using x^3-6x^2+6x=(x-2)^3-6x+8. That actually saves some work.
 
Dick said:
Try using x^3-6x^2+6x=(x-2)^3-6x+8. That actually saves some work.

Thanks. It did save a lot of work.
 

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