If a-b=1 and ab=6 then what is the value of `(a^(3)-b^(3))?`

Options

A. 21
B. 23
C. 19
D. 25

Answer

To solve the problem, we are given two equations:

1. \( a - b = 1 \)
2. \( ab = 6 \)

We need to find the value of \( a^3 - b^3 \).

### Step 1: Use the identity for \( a^3 - b^3 \)

The formula for the difference of cubes is:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

### Step 2: Substitute \( a - b \)

From the first equation, we know:
\[ a - b = 1 \]
Now we can substitute this into our formula:
\[ a^3 - b^3 = 1 \cdot (a^2 + ab + b^2) = a^2 + ab + b^2 \]

### Step 3: Find \( a^2 + b^2 \)

To find \( a^2 + b^2 \), we can use the identity:
\[ a^2 + b^2 = (a - b)^2 + 2ab \]
Substituting the known values:
\[ a^2 + b^2 = (1)^2 + 2 \cdot 6 = 1 + 12 = 13 \]

### Step 4: Substitute \( a^2 + b^2 \) into the equation

Now we can substitute \( a^2 + b^2 \) back into our expression for \( a^3 - b^3 \):
\[ a^3 - b^3 = a^2 + ab + b^2 = (a^2 + b^2) + ab = 13 + 6 = 19 \]

### Final Answer

Thus, the value of \( a^3 - b^3 \) is:
\[ \boxed{19} \]