Expand the following by using the identity
`(a-b)^(3)=a^(3)-b^(3)-3ab(a-b)`
`((a)/(11)-1)^(3)`

Answer

To expand the expression \(\left(\frac{a}{11} - 1\right)^{3}\) using the identity \((a-b)^{3} = a^{3} - b^{3} - 3ab(a-b)\), we will follow these steps:

### Step 1: Identify \(a\) and \(b\)
In our case, we can identify:
- \(a = \frac{a}{11}\)
- \(b = 1\)

### Step 2: Substitute into the identity
Using the identity, we substitute \(a\) and \(b\):
\[ \left(\frac{a}{11} - 1\right)^{3} = \left(\frac{a}{11}\right)^{3} - (1)^{3} - 3\left(\frac{a}{11}\right)(1)\left(\frac{a}{11} - 1\right) \]

### Step 3: Calculate \(a^{3}\) and \(b^{3}\)
Now we calculate \(a^{3}\) and \(b^{3}\):
- \(a^{3} = \left(\frac{a}{11}\right)^{3} = \frac{a^{3}}{11^{3}} = \frac{a^{3}}{1331}\)
- \(b^{3} = 1^{3} = 1\)

### Step 4: Substitute \(a^{3}\) and \(b^{3}\) into the equation
Now substituting these values back into the equation:
\[ \left(\frac{a}{11} - 1\right)^{3} = \frac{a^{3}}{1331} - 1 - 3\left(\frac{a}{11}\right)(1)\left(\frac{a}{11} - 1\right) \]

### Step 5: Simplify the expression
Now we simplify the term \(3\left(\frac{a}{11}\right)(1)\left(\frac{a}{11} - 1\right)\):
\[ = 3\left(\frac{a}{11}\right)\left(\frac{a}{11} - 1\right) = 3\left(\frac{a}{11}\right)\left(\frac{a - 11}{11}\right) = \frac{3a(a - 11)}{121} \]

### Step 6: Combine all parts
Now substituting this back into the equation:
\[ \left(\frac{a}{11} - 1\right)^{3} = \frac{a^{3}}{1331} - 1 - \frac{3a(a - 11)}{121} \]

### Step 7: Final expression
Thus, the final expanded expression is:
\[ \left(\frac{a}{11} - 1\right)^{3} = \frac{a^{3}}{1331} - 1 - \frac{3a(a - 11)}{121} \]