If `a + b = 8` and `ab = (32)/(3)`, then `(a^(3) + b^(3))` is equal to :

Options

A. 256
B. 128
C. 320
D. 384

Answer

To find the value of \( a^3 + b^3 \) given that \( a + b = 8 \) and \( ab = \frac{32}{3} \), we can use the formula for the sum of cubes:

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

### Step 1: Calculate \( a^2 + b^2 \)

We know that:

\[ a^2 + b^2 = (a + b)^2 - 2ab \]

Substituting the known values:

\[ a + b = 8 \quad \text{and} \quad ab = \frac{32}{3} \]

Calculating \( (a + b)^2 \):

\[ (a + b)^2 = 8^2 = 64 \]

Now substituting into the formula for \( a^2 + b^2 \):

\[ a^2 + b^2 = 64 - 2 \left(\frac{32}{3}\right) \]

Calculating \( 2 \times \frac{32}{3} \):

\[ 2 \times \frac{32}{3} = \frac{64}{3} \]

Now substituting this back:

\[ a^2 + b^2 = 64 - \frac{64}{3} \]

Finding a common denominator:

\[ 64 = \frac{192}{3} \]

Thus,

\[ a^2 + b^2 = \frac{192}{3} - \frac{64}{3} = \frac{128}{3} \]

### Step 2: Calculate \( a^2 - ab + b^2 \)

Now we can find \( a^2 - ab + b^2 \):

\[ a^2 - ab + b^2 = a^2 + b^2 - ab \]

Substituting the values we have:

\[ a^2 - ab + b^2 = \frac{128}{3} - \frac{32}{3} = \frac{128 - 32}{3} = \frac{96}{3} = 32 \]

### Step 3: Calculate \( a^3 + b^3 \)

Now we can substitute back into the formula for \( a^3 + b^3 \):

\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

Substituting the known values:

\[ a^3 + b^3 = 8 \times 32 = 256 \]

Thus, the value of \( a^3 + b^3 \) is:

\[ \boxed{256} \]