If `a + (1)/(a) = 2`, what is the value of `(a^(4) - (1)/(a^(4)))` ?

Options

A. 0
B. `(1)/(4)`
C. 1
D. 4

Answer

To solve the problem, we start with the equation given:

1. **Given Equation**:
\[ a + \frac{1}{a} = 2 \]

2. **Finding the value of \(a\)**:
We can rearrange the equation to find \(a\). Let's isolate \(a\):
\[ a + \frac{1}{a} - 2 = 0 \]
Multiplying through by \(a\) to eliminate the fraction gives:
\[ a^2 - 2a + 1 = 0 \]
This can be factored as:
\[ (a - 1)^2 = 0 \]
Therefore, we find:
\[ a - 1 = 0 \implies a = 1 \]

3. **Calculating \(a^4 - \frac{1}{a^4}\)**:
Now that we have \(a = 1\), we can substitute this value into the expression we need to evaluate:
\[ a^4 - \frac{1}{a^4} = 1^4 - \frac{1}{1^4} \]
Simplifying this gives:
\[ 1 - 1 = 0 \]

4. **Final Answer**:
Thus, the value of \(a^4 - \frac{1}{a^4}\) is:
\[ \boxed{0} \]