If `a^(2) + (1)/(a^(2))= 47 and a ne 0`, find:
`a^(3) + (1)/(a^(3))`

Answer

To solve the problem, we need to find the value of \( a^3 + \frac{1}{a^3} \) given that \( a^2 + \frac{1}{a^2} = 47 \) and \( a \neq 0 \).

### Step-by-step Solution:

1. **Start with the given equation:**
\[ a^2 + \frac{1}{a^2} = 47 \]

2. **Use the identity for \( a + \frac{1}{a} \):**
We know that:
\[ \left(a + \frac{1}{a}\right)^2 = a^2 + 2 + \frac{1}{a^2} \]
Rearranging gives us:
\[ a^2 + \frac{1}{a^2} = \left(a + \frac{1}{a}\right)^2 - 2 \]
Substituting the value we have:
\[ 47 = \left(a + \frac{1}{a}\right)^2 - 2 \]
Adding 2 to both sides:
\[ \left(a + \frac{1}{a}\right)^2 = 49 \]

3. **Take the square root:**
\[ a + \frac{1}{a} = \pm 7 \]

4. **Use the identity for \( a^3 + \frac{1}{a^3} \):**
We know that:
\[ a^3 + \frac{1}{a^3} = \left(a + \frac{1}{a}\right)^3 - 3\left(a + \frac{1}{a}\right) \]
Substituting \( a + \frac{1}{a} = 7 \) (we can ignore -7 as it will yield the same result):
\[ a^3 + \frac{1}{a^3} = 7^3 - 3 \times 7 \]

5. **Calculate \( 7^3 \) and \( 3 \times 7 \):**
\[ 7^3 = 343 \]
\[ 3 \times 7 = 21 \]

6. **Substitute and simplify:**
\[ a^3 + \frac{1}{a^3} = 343 - 21 = 322 \]

### Final Answer:
\[ a^3 + \frac{1}{a^3} = 322 \]