If a is positive and `a^2 + (1)/(a^2) = 7`, then `a^3 + (1)/(a^3)` =

Options

A. 21
B. `3sqrt7`
C. 18
D. `7sqrt7`

Answer

To solve the problem, we need to find the value of \( a^3 + \frac{1}{a^3} \) given that \( a^2 + \frac{1}{a^2} = 7 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ a^2 + \frac{1}{a^2} = 7 \]

2. **Add 2 to both sides:**
\[ a^2 + \frac{1}{a^2} + 2 = 7 + 2 \]
This simplifies to:
\[ a^2 + 2 + \frac{1}{a^2} = 9 \]

3. **Recognize the left side as a perfect square:**
\[ \left(a + \frac{1}{a}\right)^2 = 9 \]

4. **Take the square root of both sides:**
\[ a + \frac{1}{a} = \sqrt{9} \]
Since \( a \) is positive, we only consider the positive root:
\[ a + \frac{1}{a} = 3 \]

5. **Now, we need to find \( a^3 + \frac{1}{a^3} \). Use the identity:**
\[ a^3 + \frac{1}{a^3} = \left(a + \frac{1}{a}\right)^3 - 3\left(a + \frac{1}{a}\right) \]

6. **Substituting \( a + \frac{1}{a} = 3 \) into the identity:**
\[ a^3 + \frac{1}{a^3} = 3^3 - 3 \cdot 3 \]
Calculate \( 3^3 \):
\[ 3^3 = 27 \]
Now calculate \( 3 \cdot 3 \):
\[ 3 \cdot 3 = 9 \]

7. **Combine the results:**
\[ a^3 + \frac{1}{a^3} = 27 - 9 = 18 \]

### Final Answer:
\[ a^3 + \frac{1}{a^3} = 18 \]