If `a + 1/a = 3`, then the value of `a^3 + 1/(a^3)` is :

Options

A. 15
B. 18
C. 27
D. none of these

Answer

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

### Step-by-step Solution:

1. **Start with the given equation**:
\[ a + \frac{1}{a} = 3 \]

2. **Cube both sides**:
\[ \left(a + \frac{1}{a}\right)^3 = 3^3 \]
This simplifies to:
\[ a^3 + 3a\left(\frac{1}{a}\right)(a + \frac{1}{a}) + \frac{1}{a^3} = 27 \]

3. **Simplify the left side**:
The term \( 3a\left(\frac{1}{a}\right) \) simplifies to \( 3 \) (since \( a \cdot \frac{1}{a} = 1 \)). Therefore, we can rewrite the equation as:
\[ a^3 + \frac{1}{a^3} + 3(a + \frac{1}{a}) = 27 \]

4. **Substitute the value of \( a + \frac{1}{a} \)**:
Since \( a + \frac{1}{a} = 3 \), we substitute this into the equation:
\[ a^3 + \frac{1}{a^3} + 3 \cdot 3 = 27 \]
This simplifies to:
\[ a^3 + \frac{1}{a^3} + 9 = 27 \]

5. **Isolate \( a^3 + \frac{1}{a^3} \)**:
Now, we subtract 9 from both sides:
\[ a^3 + \frac{1}{a^3} = 27 - 9 \]
This gives us:
\[ a^3 + \frac{1}{a^3} = 18 \]

### Final Answer:
Thus, the value of \( a^3 + \frac{1}{a^3} \) is \( \boxed{18} \).