If x = 2 + 2 2/3 + 2 1/3 , then the value of the expression x 3- 6x 2+ 6x will be
2
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
| Concept | Description | Application in Problem |
|---|---|---|
| Substitution | Replacing an expression or variable with another equivalent one to simplify. | Letting \(a = 2^{1/3}\) to simplify the expression for \(x\). |
| Binomial Expansion | Formulas for expanding powers of binomials, like \((u+v)^3\) and \((u-v)^3\). | Used to expand \((x-2)^3\) and \((a+a^2)^3\). |
| Properties of Exponents | Rules for manipulating powers, e.g., \((b^{m/n})^k = b^{mk/n}\). | Used to calculate \(a^3\) and \(a^6\) from \(a = 2^{1/3}\). |
| Algebraic Manipulation | Rearranging equations to isolate desired terms or solve for variables. | Rearranging \(x - 2 = a + a^2\) and \(x^3 - 6x^2 + 12x - 8 = 6x - 6\). |
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.
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