If `a^(3)-b^(3)` =91 and a-b =1 what is the value of (ab)

Options

A. 27
B. 6
C. 9
D. 30

Answer

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

1. \( a^3 - b^3 = 91 \)
2. \( a - b = 1 \)

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

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

### Step 2: Substitute the value of \( a - b \)
From the second equation, we know that \( a - b = 1 \). Substitute this into the identity:
\[ a^3 - b^3 = (1)(a^2 + ab + b^2) \]
Thus, we can rewrite the first equation as:
\[ 91 = a^2 + ab + b^2 \]

### Step 3: Express \( a^2 + b^2 \) in terms of \( ab \)
We know that:
\[ a^2 + b^2 = (a - b)^2 + 2ab \]
Substituting \( a - b = 1 \):
\[ a^2 + b^2 = 1^2 + 2ab = 1 + 2ab \]

### Step 4: Substitute \( a^2 + b^2 \) into the equation
Now substitute \( a^2 + b^2 \) into the equation we derived from the difference of cubes:
\[ 91 = (1 + 2ab) + ab \]
This simplifies to:
\[ 91 = 1 + 3ab \]

### Step 5: Solve for \( ab \)
Now, isolate \( ab \):
\[ 91 - 1 = 3ab \]
\[ 90 = 3ab \]
\[ ab = \frac{90}{3} = 30 \]

### Conclusion
Thus, the value of \( ab \) is:
\[ \boxed{30} \]