If `x=4+(1)/(a)` and `y=2-(1)/(2a)`, what is the value of `(x^(3)+6x^(2)y+12xy^(2)+8y^(3))`?

Answer

To solve the problem, we need to evaluate the expression \(x^3 + 6x^2y + 12xy^2 + 8y^3\) given the values of \(x\) and \(y\):

1. **Given Values**:
\[ x = 4 + \frac{1}{a} \]
\[ y = 2 - \frac{1}{2a} \]

2. **Recognizing the Expression**:
The expression \(x^3 + 6x^2y + 12xy^2 + 8y^3\) resembles the expansion of \((x + 2y)^3\). Recall the identity:
\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]
Here, we can set \(a = x\) and \(b = 2y\).

3. **Identifying \(2y\)**:
First, calculate \(2y\):
\[ 2y = 2 \left(2 - \frac{1}{2a}\right) = 4 - \frac{1}{a} \]

4. **Combining \(x\) and \(2y\)**:
Now, we can find \(x + 2y\):
\[ x + 2y = \left(4 + \frac{1}{a}\right) + \left(4 - \frac{1}{a}\right) \]
Simplifying this:
\[ x + 2y = 4 + \frac{1}{a} + 4 - \frac{1}{a} = 8 \]

5. **Calculating the Cube**:
Now, substitute \(x + 2y\) into the cube:
\[ (x + 2y)^3 = 8^3 = 512 \]

6. **Final Result**:
Therefore, the value of the expression \(x^3 + 6x^2y + 12xy^2 + 8y^3\) is:
\[ \boxed{512} \]