If `a^3 + b^3 = 105`, and (a + b) = 5 then what is the value of ab?

Options

A. 21
B. `4//3`
C. 7
D. `3//4`

Answer

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

1. \( a^3 + b^3 = 105 \)
2. \( a + b = 5 \)

We want to find the value of \( ab \).

### Step 1: Use the identity for the sum of cubes
We can use the identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

### Step 2: Rewrite \( a^2 - ab + b^2 \)
We know that:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting \( a + b = 5 \):
\[ a^2 + b^2 = 5^2 - 2ab = 25 - 2ab \]
Now, we can rewrite \( a^2 - ab + b^2 \) as:
\[ a^2 - ab + b^2 = (a^2 + b^2) - ab = (25 - 2ab) - ab = 25 - 3ab \]

### Step 3: Substitute into the sum of cubes identity
Now substituting back into the sum of cubes identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
This gives us:
\[ 105 = 5(25 - 3ab) \]

### Step 4: Solve for \( ab \)
Now, we can simplify:
\[ 105 = 125 - 15ab \]
Rearranging gives:
\[ 15ab = 125 - 105 \]
\[ 15ab = 20 \]
Now, dividing both sides by 15:
\[ ab = \frac{20}{15} = \frac{4}{3} \]

### Final Answer
Thus, the value of \( ab \) is:
\[ \boxed{\frac{4}{3}} \]