Determine the value of `(y-1/y)^(2)` when `y^(4) + 1/y^(4) = 34`.

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to determine the value of \((y - \frac{1}{y})^2\) given that \(y^4 + \frac{1}{y^4} = 34\).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ y^4 + \frac{1}{y^4} = 34 \]

2. **Use the identity for \(y^4 + \frac{1}{y^4}\):**
We can relate \(y^4 + \frac{1}{y^4}\) to \(y^2 + \frac{1}{y^2}\) using the identity:
\[ y^4 + \frac{1}{y^4} = \left(y^2 + \frac{1}{y^2}\right)^2 - 2 \]
Let \(x = y^2 + \frac{1}{y^2}\). Then we can rewrite the equation as:
\[ x^2 - 2 = 34 \]

3. **Solve for \(x^2\):**
\[ x^2 = 34 + 2 = 36 \]

4. **Take the square root of both sides:**
\[ x = \sqrt{36} = 6 \]
(We take the positive root since \(y^2 + \frac{1}{y^2}\) is always positive.)

5. **Now relate \(x\) back to \((y - \frac{1}{y})^2\):**
We use the identity:
\[ y^2 + \frac{1}{y^2} = \left(y - \frac{1}{y}\right)^2 + 2 \]
Substituting \(x\):
\[ 6 = \left(y - \frac{1}{y}\right)^2 + 2 \]

6. **Solve for \((y - \frac{1}{y})^2\):**
\[ \left(y - \frac{1}{y}\right)^2 = 6 - 2 = 4 \]

### Final Answer:
Thus, the value of \((y - \frac{1}{y})^2\) is:
\[ \boxed{4} \]