Understanding the Problem and Goal

The question asks us to evaluate the value of a specific cubic expression, \(8x^3 + \frac{1}{x^3}\), given a conditional equation involving \(x\), which is \(x + \frac{1}{2x} = 3\). To solve this, we need to manipulate the given equation to relate it to the expression we need to evaluate, likely using algebraic identities.

Step-by-Step Solution

We are given the equation: \(x + \frac{1}{2x} = 3\)

And we need to find the value of the expression: \(8x^3 + \frac{1}{x^3}\)

Notice that the expression \(8x^3 + \frac{1}{x^3}\) can be written as \((2x)^3 + (\frac{1}{x})^3\). This looks like the sum of two cubes.

Let's examine the given equation. If we multiply the entire equation by 2, we might get a term involving \(2x\):

\(2 \times (x + \frac{1}{2x}) = 2 \times 3\)

Distributing the 2 on the left side gives:

\(2x + 2 \times \frac{1}{2x} = 6\)

\(2x + \frac{2}{2x} = 6\)

\(2x + \frac{1}{x} = 6\)

This result, \(2x + \frac{1}{x} = 6\), is very useful because it relates directly to the terms in the cubic expression \((2x)^3 + (\frac{1}{x})^3\). Let's use an algebraic identity for the sum of cubes.

We know the identity: \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

Rearranging this identity to solve for \(a^3 + b^3\):

\(a^3 + b^3 = (a+b)^3 - 3a^2b - 3ab^2\)

\(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\)

In our case, we want to evaluate \((2x)^3 + (\frac{1}{x})^3\). Let's set \(a = 2x\) and \(b = \frac{1}{x}\).

Using the identity \(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\):

\((2x)^3 + (\frac{1}{x})^3 = (2x + \frac{1}{x})^3 - 3(2x)(\frac{1}{x})(2x + \frac{1}{x})\)

From our manipulation of the given equation, we found that \(2x + \frac{1}{x} = 6\).

Let's find the product \(ab\):

\(ab = (2x)(\frac{1}{x}) = 2 \times \frac{x}{x} = 2 \times 1 = 2\)

Now, substitute the values \(2x + \frac{1}{x} = 6\) and \(ab = 2\) into the expanded identity:

\((2x)^3 + (\frac{1}{x})^3 = (6)^3 - 3(2)(6)\)

Calculate the terms:

\((6)^3 = 6 \times 6 \times 6 = 36 \times 6 = 216\)

\(3(2)(6) = 6 \times 6 = 36\)

So, the expression becomes:

\(8x^3 + \frac{1}{x^3} = 216 - 36\)

\(8x^3 + \frac{1}{x^3} = 180\)

Therefore, the value of the expression \(8x^3 + \frac{1}{x^3}\) is 180.

Summary of Calculation Steps

1. Start with the given equation \(x + \frac{1}{2x} = 3\).
2. Identify the expression to evaluate: \(8x^3 + \frac{1}{x^3}\).
3. Multiply the given equation by 2 to get \(2x + \frac{1}{x} = 6\).
4. Recognize the expression as a sum of cubes: \((2x)^3 + (\frac{1}{x})^3\).
5. Use the algebraic identity \(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\) with \(a = 2x\) and \(b = \frac{1}{x}\).
6. Substitute \(a+b = 2x + \frac{1}{x} = 6\) and \(ab = (2x)(\frac{1}{x}) = 2\) into the identity.
7. Calculate \((6)^3 - 3(2)(6) = 216 - 36 = 180\).

Revision Table: Key Concepts for Evaluating Expressions

Additional Information: Exploring Related Algebraic Identities

Problems like this often rely on recognizing patterns and applying the correct algebraic identities. Here are a few other common identities that might be useful in similar problems:

• \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 = a^3 - b^3 - 3ab(a-b)\)
• Difference of Cubes: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)
• Square of a Sum: \((a+b)^2 = a^2 + 2ab + b^2\)
• Square of a Difference: \((a-b)^2 = a^2 - 2ab + b^2\)

Being familiar with these identities and how to rearrange them can simplify complex algebraic expressions and equations. In this problem, rewriting \(8x^3 + \frac{1}{x^3}\) as \((2x)^3 + (\frac{1}{x})^3\) was key to identifying the correct identity to use based on the derived term \(2x + \frac{1}{x}\).