If `a + (1)/(a) = 3`, then `(a^(4) + (1)/(a^(4)))` is equal to :

Options

A. A)77
B. B)47
C. C)81
D. D)27

Answer

To solve the equation \( a + \frac{1}{a} = 3 \) and find \( a^4 + \frac{1}{a^4} \), we can follow these steps:

### Step 1: Square both sides of the equation
Starting with the given equation:
\[ a + \frac{1}{a} = 3 \]
We square both sides:
\[ \left( a + \frac{1}{a} \right)^2 = 3^2 \]
This simplifies to:
\[ a^2 + 2 + \frac{1}{a^2} = 9 \]

### Step 2: Rearrange to find \( a^2 + \frac{1}{a^2} \)
Now, we can rearrange the equation to isolate \( a^2 + \frac{1}{a^2} \):
\[ a^2 + \frac{1}{a^2} = 9 - 2 \]
Thus, we have:
\[ a^2 + \frac{1}{a^2} = 7 \]

### Step 3: Square again to find \( a^4 + \frac{1}{a^4} \)
Next, we square \( a^2 + \frac{1}{a^2} \):
\[ \left( a^2 + \frac{1}{a^2} \right)^2 = 7^2 \]
This expands to:
\[ a^4 + 2 + \frac{1}{a^4} = 49 \]

### Step 4: Rearrange to find \( a^4 + \frac{1}{a^4} \)
Now, we rearrange to isolate \( a^4 + \frac{1}{a^4} \):
\[ a^4 + \frac{1}{a^4} = 49 - 2 \]
Thus, we find:
\[ a^4 + \frac{1}{a^4} = 47 \]

### Final Answer
Therefore, the value of \( a^4 + \frac{1}{a^4} \) is:
\[ \boxed{47} \]