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

Answer

To solve the problem, we start with the equation given:

**Step 1:** Given the equation:
\[ a - \frac{1}{a} = 3 \]

**Step 2:** We want to find \( a^3 - \frac{1}{a^3} \). We can use the identity:
\[ a^3 - \frac{1}{a^3} = \left( a - \frac{1}{a} \right) \left( a^2 + 1 + \frac{1}{a^2} \right) \]

**Step 3:** First, we need to find \( a^2 + \frac{1}{a^2} \). We can use the square of the original equation:
\[ \left( a - \frac{1}{a} \right)^2 = a^2 - 2 + \frac{1}{a^2} \]
Substituting the value:
\[ 3^2 = a^2 - 2 + \frac{1}{a^2} \]
\[ 9 = a^2 - 2 + \frac{1}{a^2} \]
\[ a^2 + \frac{1}{a^2} = 9 + 2 = 11 \]

**Step 4:** Now we can substitute \( a^2 + \frac{1}{a^2} \) back into our identity:
\[ a^3 - \frac{1}{a^3} = \left( a - \frac{1}{a} \right) \left( a^2 + 1 + \frac{1}{a^2} \right) \]
Substituting the known values:
\[ a^3 - \frac{1}{a^3} = 3 \left( 11 + 1 \right) = 3 \times 12 = 36 \]

Thus, the final answer is:
\[ \boxed{36} \]