Evaluating the Algebraic Expression \(x^3 - 6x^2 + 6x\)

We are given an expression for \(x\) and asked to find the value of a related cubic expression. The given expression for \(x\) is \(x = 2 + 2^{2/3} + 2^{1/3}\). We need to find the value of \(x^3 - 6x^2 + 6x\).

Simplifying the Expression for \(x\) using Substitution

Let's simplify the expression for \(x\) by using a substitution. Notice the terms involving the cube root of 2. Let \(a = 2^{1/3}\). Then, \(a^2 = (2^{1/3})^2 = 2^{2/3}\).

Substituting these into the expression for \(x\), we get:

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

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

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

Cubing Both Sides of the Equation

To introduce the cubic terms of \(x\) and eliminate the fractional exponents involving \(a\), we can cube both sides of the equation \(x - 2 = a + a^2\).

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

Expanding the Left Side: \((x-2)^3\)

We use the binomial expansion formula \((u-v)^3 = u^3 - 3u^2v + 3uv^2 - v^3\) with \(u=x\) and \(v=2\).

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

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

Expanding the Right Side: \((a+a^2)^3\)

We use the binomial expansion formula \((u+v)^3 = u^3 + v^3 + 3uv(u+v)\) with \(u=a\) and \(v=a^2\).

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

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

Now, substitute the values based on \(a = 2^{1/3}\):

• \(a^3 = (2^{1/3})^3 = 2\)
• \(a^6 = (2^{1/3})^6 = 2^{(1/3) \times 6} = 2^2 = 4\)
• Recall that \(a + a^2 = x - 2\) from our initial rearrangement.

Substitute these values into the expanded right side:

\((a + a^2)^3 = 2 + 4 + 3(2)(x - 2)\)

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

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

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

Equating the Expanded Sides

Now, we set the expanded left side equal to the expanded right side:

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

Solving for the Expression \(x^3 - 6x^2 + 6x\)

We want to find the value of \(x^3 - 6x^2 + 6x\). We can rearrange the equation obtained in the previous step to isolate this expression:

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

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

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

Revision Table: Algebraic Evaluation and Expansion

Additional Information: Working with Rational Exponents

Rational exponents are exponents that are fractions. They connect roots and powers. A term like \(b^{m/n}\) can be understood as the \(n\)-th root of \(b\) raised to the power of \(m\), i.e., \(b^{m/n} = \sqrt[n]{b^m} = (\sqrt[n]{b})^m\).

Key properties used in this problem include:

• \(b^{1/n} = \sqrt[n]{b}\) (e.g., \(2^{1/3} = \sqrt[3]{2}\))
• \((b^p)^q = b^{pq}\) (e.g., \((2^{1/3})^3 = 2^{(1/3) \times 3} = 2^1 = 2\), and \((2^{1/3})^6 = 2^{(1/3) \times 6} = 2^2 = 4\))

Understanding these properties is crucial for simplifying expressions involving roots and fractional powers, which then allows for easier algebraic manipulation and evaluation of polynomial expressions.