If `a- (1)/( a) = 3`, find `a^(3) - (1)/(a^3)`

Answer

To solve the equation \( a - \frac{1}{a} = 3 \) and find \( a^3 - \frac{1}{a^3} \), we can use the identity related to cubes.

### Step-by-step Solution:

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

2. **Cube both sides of the equation**:
Using the identity \( (x - y)^3 = x^3 - y^3 - 3xy(x - y) \), we can let \( x = a \) and \( y = \frac{1}{a} \). Therefore:
\[ (a - \frac{1}{a})^3 = a^3 - \left(\frac{1}{a}\right)^3 - 3 \cdot a \cdot \frac{1}{a} \cdot (a - \frac{1}{a}) \]

3. **Substituting the values**:
We know \( a - \frac{1}{a} = 3 \). Thus:
\[ (3)^3 = a^3 - \frac{1}{a^3} - 3 \cdot 1 \cdot 3 \]

4. **Calculate \( 3^3 \)**:
\[ 27 = a^3 - \frac{1}{a^3} - 9 \]

5. **Rearranging the equation**:
\[ a^3 - \frac{1}{a^3} = 27 + 9 \]

6. **Final calculation**:
\[ a^3 - \frac{1}{a^3} = 36 \]

### Final Answer:
\[ a^3 - \frac{1}{a^3} = 36 \]