Evaluating Algebraic Expressions

We are given the value of \(x\) and asked to find the value of a specific algebraic expression involving \(x\).

The given information is:

• \(x = 2 + 2^{2/3} + 2^{1/3}\)
• We need to find the value of \(x^3 - 6x^2 + 6x\)

Let's simplify the expression for \(x\). Notice the terms \(2^{2/3}\) and \(2^{1/3}\). These are related to cube roots of 2.

Let \(a = 2^{1/3}\).

Then \(a^2 = (2^{1/3})^2 = 2^{2/3}\).

Also, \(a^3 = (2^{1/3})^3 = 2\).

Substituting this into the expression for \(x\):

\(x = 2 + a^2 + a\)

We can rearrange this equation to isolate the terms involving \(a\):

\(x - 2 = a^2 + a\)

Now, let's consider the expression we need to evaluate: \(x^3 - 6x^2 + 6x\).

This expression looks related to the expansion of \((x-2)^3\). Let's expand \((x-2)^3\):

\((x-2)^3 = x^3 - 3(x^2)(2) + 3(x)(2^2) - 2^3\)

\((x-2)^3 = x^3 - 6x^2 + 12x - 8\)

The expression we want to evaluate is \(x^3 - 6x^2 + 6x\).

We can rewrite this as:

\(x^3 - 6x^2 + 6x = (x^3 - 6x^2 + 12x - 8) - 6x + 8\)

Substituting the expansion of \((x-2)^3\):

\(x^3 - 6x^2 + 6x = (x-2)^3 - 6x + 8\)

This doesn't seem to directly simplify things yet. Let's go back to the equation \((x-2) = a^2 + a\) and cube both sides.

\((x-2)^3 = (a^2 + a)^3\)

We can factor out \(a\) from the right side: \(a^2 + a = a(a+1)\).

So, \((a^2 + a)^3 = (a(a+1))^3 = a^3 (a+1)^3\).

We know \(a^3 = 2\).

Now let's expand \((a+1)^3\):

\((a+1)^3 = a^3 + 3a^2 + 3a + 1\)

Substitute this back into the expression for \((x-2)^3\):

\((x-2)^3 = a^3 (a^3 + 3a^2 + 3a + 1)\)

\((x-2)^3 = 2 (2 + 3a^2 + 3a + 1)\)

\((x-2)^3 = 2 (3 + 3a^2 + 3a)\)

\((x-2)^3 = 6 + 6a^2 + 6a\)

We can factor out 6 from the last two terms: \(6a^2 + 6a = 6(a^2 + a)\).

From our earlier step, we know that \(a^2 + a = x - 2\).

Substitute \(a^2 + a = x - 2\) back into the equation for \((x-2)^3\):

\((x-2)^3 = 6 + 6(a^2 + a)\)

\((x-2)^3 = 6 + 6(x - 2)\)

\((x-2)^3 = 6 + 6x - 12\)

\((x-2)^3 = 6x - 6\)

Now, expand the left side \((x-2)^3\) again:

\(x^3 - 6x^2 + 12x - 8 = 6x - 6\)

We want to find the value of \(x^3 - 6x^2 + 6x\). Let's rearrange the terms in the equation we just derived to get this expression:

Move the \(6x\) term from the right side to the left side, and the constant term \(-8\) from the left side to the right side.

\(x^3 - 6x^2 + 12x - 6x = 8 - 6\)

\(x^3 - 6x^2 + 6x = 2\)

So, the value of the expression \(x^3 - 6x^2 + 6x\) is 2.

Revision Table: Key Steps to Evaluate the Expression

Additional Information on Algebraic Manipulation

This problem demonstrates a common technique in algebra: recognizing how a given expression relates to powers of a modified variable, like \((x-k)\). In this case, recognizing that the expression \(x^3 - 6x^2 + 6x\) is similar to the expansion of \((x-2)^3 = x^3 - 6x^2 + 12x - 8\) was helpful.

The substitution \(a = 2^{1/3}\) simplified the initial expression for \(x\) and allowed us to work with a simpler relationship between \(x\) and \(a\). The property \(a^3 = 2\) was crucial in simplifying the cubed expression.

Using the relationship \(x-2 = a^2+a\) repeatedly during the expansion of \((x-2)^3\) was key to solving this problem efficiently. It allowed us to replace terms involving \(a\) with terms involving \(x\), ultimately leading to an equation solely in terms of \(x\) which could then be rearranged to find the value of the target expression.