If `a^(3) + b^(3) = 416` and `a + b = 16`, then `(a-b)^(2) + ab` is equal to :

Options

A. 32
B. 22
C. 24
D. 26

Answer

To solve the problem, we need to find the value of \((a-b)^2 + ab\) given that \(a^3 + b^3 = 416\) and \(a + b = 16\).

### Step-by-Step Solution:

1. **Use the identity for the sum of cubes**:
The formula for the sum of cubes is:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
We know \(a + b = 16\) and \(a^3 + b^3 = 416\). Plugging these values into the identity gives:
\[ 416 = 16(a^2 - ab + b^2) \]

2. **Simplify the equation**:
Divide both sides by 16:
\[ a^2 - ab + b^2 = \frac{416}{16} = 26 \]

3. **Use the identity for \(a^2 + b^2\)**:
We can express \(a^2 + b^2\) in terms of \(a + b\) and \(ab\):
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting \(a + b = 16\):
\[ a^2 + b^2 = 16^2 - 2ab = 256 - 2ab \]

4. **Substitute \(a^2 + b^2\) into the previous equation**:
Now we substitute \(a^2 + b^2\) into the equation we derived earlier:
\[ (256 - 2ab) - ab = 26 \]
This simplifies to:
\[ 256 - 3ab = 26 \]

5. **Solve for \(ab\)**:
Rearranging the equation gives:
\[ 3ab = 256 - 26 = 230 \]
Therefore:
\[ ab = \frac{230}{3} \approx 76.67 \]

6. **Now calculate \((a-b)^2 + ab\)**:
We can express \((a-b)^2\) using \(a^2 + b^2\) and \(ab\):
\[ (a-b)^2 = a^2 + b^2 - 2ab \]
Substituting \(a^2 + b^2 = 256 - 2ab\) into this gives:
\[ (a-b)^2 = (256 - 2ab) - 2ab = 256 - 4ab \]
Now substituting \(ab = \frac{230}{3}\):
\[ (a-b)^2 = 256 - 4 \times \frac{230}{3} = 256 - \frac{920}{3} = \frac{768 - 920}{3} = \frac{-152}{3} \]

7. **Finally, calculate \((a-b)^2 + ab\)**:
\[ (a-b)^2 + ab = \frac{-152}{3} + \frac{230}{3} = \frac{78}{3} = 26 \]

Thus, the final answer is:
\[ \boxed{26} \]