If `a^(4)+(1)/(a^(4))=47` then find the value of `a+(1)/(a)`.

Answer

To solve the problem, we need to find the value of \( a + \frac{1}{a} \) given that \( a^4 + \frac{1}{a^4} = 47 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ a^4 + \frac{1}{a^4} = 47 \]

2. **Use the identity for \( a^4 + \frac{1}{a^4} \):**
We can express \( a^4 + \frac{1}{a^4} \) in terms of \( a^2 + \frac{1}{a^2} \):
\[ a^4 + \frac{1}{a^4} = \left( a^2 + \frac{1}{a^2} \right)^2 - 2 \]
Let \( x = a^2 + \frac{1}{a^2} \). Then we can rewrite the equation as:
\[ x^2 - 2 = 47 \]

3. **Solve for \( x^2 \):**
\[ x^2 = 47 + 2 = 49 \]

4. **Take the square root to find \( x \):**
\[ x = \sqrt{49} = 7 \]
Since \( a^2 + \frac{1}{a^2} \) is positive, we take the positive root.

5. **Now, express \( a + \frac{1}{a} \) in terms of \( x \):**
We use the identity:
\[ a^2 + \frac{1}{a^2} = \left( a + \frac{1}{a} \right)^2 - 2 \]
Let \( y = a + \frac{1}{a} \). Then we have:
\[ x = y^2 - 2 \]

6. **Substitute \( x \) into the equation:**
\[ 7 = y^2 - 2 \]

7. **Solve for \( y^2 \):**
\[ y^2 = 7 + 2 = 9 \]

8. **Take the square root to find \( y \):**
\[ y = \sqrt{9} = 3 \]
Again, since \( a + \frac{1}{a} \) is positive, we take the positive root.

### Final Answer:
\[ a + \frac{1}{a} = 3 \]