If ` a^(4) + 1 =[ (a^(2))/( b^(2))] ( 4 b ^(2) - b ^(4) - 1)` then what is the value of ` a ^(4) + b^(4)` ?

Options

A. 2
B. 16
C. 32
D. 64

Answer

To solve the equation \( a^4 + 1 = \left( \frac{a^2}{b^2} \right) (4b^2 - b^4 - 1) \) and find the value of \( a^4 + b^4 \), we can follow these steps:

### Step-by-Step Solution:

1. **Substitute a value for b**:
We can simplify the problem by assuming a value for \( b \). Let's set \( b = 1 \).

\[ a^4 + 1 = \left( \frac{a^2}{1^2} \right) (4 \cdot 1^2 - 1^4 - 1) \]

This simplifies to:

\[ a^4 + 1 = a^2 (4 - 1 - 1) \]

Which further simplifies to:

\[ a^4 + 1 = a^2 \cdot 2 \]

2. **Rearranging the equation**:
Now, we can rearrange the equation:

\[ a^4 - 2a^2 + 1 = 0 \]

3. **Factoring the equation**:
The equation can be factored as:

\[ (a^2 - 1)^2 = 0 \]

4. **Finding the values of a**:
Setting the factor equal to zero gives us:

\[ a^2 - 1 = 0 \]

Therefore,

\[ a^2 = 1 \implies a = 1 \text{ or } a = -1 \]

5. **Calculating \( a^4 + b^4 \)**:
Since we assumed \( b = 1 \), we can now find \( a^4 + b^4 \):

- If \( a = 1 \):
\[ a^4 + b^4 = 1^4 + 1^4 = 1 + 1 = 2 \]

- If \( a = -1 \):
\[ a^4 + b^4 = (-1)^4 + 1^4 = 1 + 1 = 2 \]

In both cases, we find that:

\[ a^4 + b^4 = 2 \]

### Final Answer:
Thus, the value of \( a^4 + b^4 \) is \( \boxed{2} \).