If `a^(3) - b^(3)= 210 and a-b=5 , ` then `(a+b)^(2)-ab` is equal to :

Options

A. 42
B. 32
C. 38
D. 52

Answer

To solve the problem, we need to find the value of \((a+b)^2 - ab\) given that \(a^3 - b^3 = 210\) and \(a - b = 5\).

### Step 1: Use the identity for the difference of cubes
We know the identity for the difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Given \(a^3 - b^3 = 210\) and \(a - b = 5\), we can substitute these values into the identity.

### Step 2: Substitute the known values
Substituting \(a - b = 5\) into the identity gives:
\[ 210 = 5(a^2 + ab + b^2) \]

### Step 3: Simplify the equation
Dividing both sides by 5:
\[ a^2 + ab + b^2 = \frac{210}{5} = 42 \]

### Step 4: Use the square of a sum identity
We know that:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
We can express \(a^2 + b^2\) in terms of \(a + b\) and \(ab\):
\[ a^2 + b^2 = (a + b)^2 - 2ab \]

### Step 5: Substitute into the equation
Now, we can substitute \(a^2 + b^2\) into the equation we derived:
\[ a^2 + ab + b^2 = (a + b)^2 - 2ab + ab = (a + b)^2 - ab \]
So we have:
\[ (a + b)^2 - ab = 42 \]

### Step 6: Conclusion
Thus, the value of \((a + b)^2 - ab\) is:
\[ \boxed{42} \]