If `a-b = 6 and ab = 16,` then `a ^(3) - b ^(3)` is

Options

A. 504
B. 702
C. 725
D. 511

Answer

To solve the problem where \( a - b = 6 \) and \( ab = 16 \), we need to find \( a^3 - b^3 \).

### Step-by-Step Solution:

1. **Use the formula for the difference of cubes**:
The formula for the difference of cubes is:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
We already know \( a - b = 6 \).

2. **Calculate \( a^2 + ab + b^2 \)**:
To use the formula, we need to find \( a^2 + ab + b^2 \). We can express \( a^2 + b^2 \) in terms of \( (a - b)^2 \) and \( ab \):
\[ a^2 + b^2 = (a - b)^2 + 2ab \]
Substitute the known values:
\[ a^2 + b^2 = (6)^2 + 2 \times 16 = 36 + 32 = 68 \]
Now, we can find \( a^2 + ab + b^2 \):
\[ a^2 + ab + b^2 = a^2 + b^2 + ab = 68 + 16 = 84 \]

3. **Substitute back into the difference of cubes formula**:
Now we can substitute back into the difference of cubes formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) = 6 \times 84 \]

4. **Calculate the final result**:
\[ a^3 - b^3 = 504 \]

### Final Answer:
Thus, \( a^3 - b^3 = 504 \).

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