If ` ab ( a + b ) = 1 ` then what is the value of ` ( 1) / ( a^( 3) b ^(3)) - a^(3) - b^(3) ` ?

Options

A. `-1`
B. 1
C. 3
D. `-3`

Answer

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

1. **Given:**
\[ ab(a + b) = 1 \]

2. **Rearranging the equation:**
We can express \( ab \) in terms of \( a + b \):
\[ ab = \frac{1}{a + b} \]

3. **Cubing both sides:**
We will cube the expression \( a + b \):
\[ (a + b)^3 = a^3 + b^3 + 3ab(a + b) \]

4. **Substituting \( ab \):**
Substitute \( ab = \frac{1}{a + b} \) into the equation:
\[ (a + b)^3 = a^3 + b^3 + 3\left(\frac{1}{a + b}\right)(a + b) \]

5. **Simplifying:**
The \( (a + b) \) terms cancel out:
\[ (a + b)^3 = a^3 + b^3 + 3 \]

6. **Rearranging to find \( a^3 + b^3 \):**
We can rearrange this to isolate \( a^3 + b^3 \):
\[ a^3 + b^3 = (a + b)^3 - 3 \]

7. **Finding \( \frac{1}{a^3b^3} - a^3 - b^3 \):**
We need to find the value of:
\[ \frac{1}{a^3b^3} - (a^3 + b^3) \]
We know that \( a^3b^3 = (ab)^3 \). Since \( ab = \frac{1}{a + b} \):
\[ a^3b^3 = \left(\frac{1}{a + b}\right)^3 \]

8. **Substituting into the expression:**
Thus, we have:
\[ \frac{1}{a^3b^3} = (a + b)^3 \]
Therefore:
\[ \frac{1}{a^3b^3} - (a^3 + b^3) = (a + b)^3 - ((a + b)^3 - 3) \]

9. **Simplifying further:**
This simplifies to:
\[ \frac{1}{a^3b^3} - (a^3 + b^3) = 3 \]

10. **Final answer:**
Thus, the value is:
\[ \boxed{3} \]