Factorise:` (1)/(64) a ^(3) - (1)/(16) a ^(2) b + (1)/(12) ab ^(2) - (1)/(27) b ^(3)`

Answer

To factor the expression \( \frac{1}{64} a^3 - \frac{1}{16} a^2 b + \frac{1}{12} ab^2 - \frac{1}{27} b^3 \), we can follow these steps:

### Step 1: Rewrite the coefficients
First, we can rewrite the coefficients in a more manageable form:
\[ \frac{1}{64} = \left(\frac{1}{4}\right)^3, \quad \frac{1}{16} = \left(\frac{1}{4}\right)^2, \quad \frac{1}{12} = \frac{1}{12}, \quad \frac{1}{27} = \left(\frac{1}{3}\right)^3 \]
Thus, we can rewrite the expression as:
\[ \left(\frac{a}{4}\right)^3 - \left(\frac{a}{4}\right)^2 b + \frac{1}{12} ab^2 - \left(\frac{b}{3}\right)^3 \]

### Step 2: Identify the structure
We notice that the expression resembles the form of \( A^3 - B^3 - 3A^2B + 3AB^2 \). We can set:
\[ A = \frac{a}{4}, \quad B = \frac{b}{3} \]
This allows us to rewrite the expression as:
\[ A^3 - 3A^2B + 3AB^2 - B^3 \]

### Step 3: Apply the formula
The expression can be factored using the identity:
\[ A^3 - B^3 - 3AB(A - B) = (A - B)(A^2 + AB + B^2) \]
Thus, we can factor our expression as:
\[ \left(\frac{a}{4} - \frac{b}{3}\right)\left(\left(\frac{a}{4}\right)^2 + \left(\frac{a}{4}\right)\left(\frac{b}{3}\right) + \left(\frac{b}{3}\right)^2\right) \]

### Step 4: Simplify the factors
Now we simplify the factors:
1. The first factor:
\[ \frac{a}{4} - \frac{b}{3} = \frac{3a - 4b}{12} \]
2. The second factor:
\[ \left(\frac{a}{4}\right)^2 + \left(\frac{a}{4}\right)\left(\frac{b}{3}\right) + \left(\frac{b}{3}\right)^2 = \frac{a^2}{16} + \frac{ab}{12} + \frac{b^2}{9} \]

### Final Result
Thus, the completely factored form of the expression is:
\[ \frac{1}{12} (3a - 4b) \left(\frac{a^2}{16} + \frac{ab}{12} + \frac{b^2}{9}\right) \]