Simplify using following identiy:
`(a +- b) (a^(2) ab + b^(2)) = a^(3) +- b^(3)`
`((a)/(3)-3b) ((a^(2))/(9) + ab + 9b^(2))`

Answer

To simplify the expression \(\left(\frac{a}{3} - 3b\right) \left(\frac{a^2}{9} + ab + 9b^2\right)\) using the identity \((a - b)(a^2 + ab + b^2) = a^3 - b^3\), we will follow these steps:

### Step 1: Identify \(a\) and \(b\)
In our case, we can identify:
- \(a = \frac{a}{3}\)
- \(b = 3b\)

### Step 2: Rewrite the expression using the identity
According to the identity, we can rewrite the expression:
\[ \left(\frac{a}{3} - 3b\right) \left(\frac{a^2}{9} + ab + 9b^2\right) = \left(\frac{a}{3}\right)^3 - (3b)^3 \]

### Step 3: Calculate \(\left(\frac{a}{3}\right)^3\)
Calculating \(\left(\frac{a}{3}\right)^3\):
\[ \left(\frac{a}{3}\right)^3 = \frac{a^3}{27} \]

### Step 4: Calculate \((3b)^3\)
Calculating \((3b)^3\):
\[ (3b)^3 = 27b^3 \]

### Step 5: Substitute back into the identity
Now substituting back into the identity:
\[ \left(\frac{a}{3} - 3b\right) \left(\frac{a^2}{9} + ab + 9b^2\right) = \frac{a^3}{27} - 27b^3 \]

### Final Answer
Thus, the simplified form of the given expression is:
\[ \frac{a^3}{27} - 27b^3 \]

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