If `a^(3) + b^(3) = 152` and `a + b = 8` then what is the value of ab?

Options

A. `2`
B. `11`
C. `-10`
D. `15`

Answer

To solve the problem, we need to find the value of \( ab \) given the equations \( a^3 + b^3 = 152 \) and \( a + b = 8 \).

### Step-by-Step Solution:

**Step 1:** Use the identity for the sum of cubes.
The identity for the sum of cubes states:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
We can also express \( a^2 - ab + b^2 \) in terms of \( a + b \) and \( ab \).

**Step 2:** Rewrite \( a^2 - ab + b^2 \).
We know that:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Thus,
\[ a^2 - ab + b^2 = (a^2 + b^2) - ab = (a + b)^2 - 3ab \]

**Step 3:** Substitute known values into the equation.
Substituting \( a + b = 8 \) into the equation gives:
\[ a^3 + b^3 = (8)((8)^2 - 3ab) \]
This simplifies to:
\[ a^3 + b^3 = 8(64 - 3ab) \]
\[ a^3 + b^3 = 512 - 24ab \]

**Step 4:** Set the equations equal to each other.
We know from the problem statement that \( a^3 + b^3 = 152 \). Therefore, we can set the two expressions for \( a^3 + b^3 \) equal to each other:
\[ 512 - 24ab = 152 \]

**Step 5:** Solve for \( ab \).
Rearranging the equation gives:
\[ 512 - 152 = 24ab \]
\[ 360 = 24ab \]
Now, divide both sides by 24:
\[ ab = \frac{360}{24} = 15 \]

### Final Answer:
The value of \( ab \) is \( 15 \).

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