Simplify : `(3x^(2))^(-3)xx(x^(9))^((2)/(3))`

Answer

To simplify the expression `(3x^(2))^(-3) * (x^(9))^(2/3)`, we can follow these steps:

### Step 1: Apply the Power of a Product Rule
We start with the expression:
\[ (3x^2)^{-3} \cdot (x^9)^{\frac{2}{3}} \]
We can separate the components of the first term:
\[ = 3^{-3} \cdot (x^2)^{-3} \cdot (x^9)^{\frac{2}{3}} \]

### Step 2: Simplify Each Component
Now we simplify each part:
\[ 3^{-3} = \frac{1}{3^3} \quad \text{and} \quad (x^2)^{-3} = x^{-6} \quad \text{and} \quad (x^9)^{\frac{2}{3}} = x^{6} \]
So we rewrite the expression:
\[ = \frac{1}{3^3} \cdot x^{-6} \cdot x^{6} \]

### Step 3: Combine the x Terms
Using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\):
\[ x^{-6} \cdot x^{6} = x^{-6 + 6} = x^{0} \]
Thus, the expression simplifies to:
\[ = \frac{1}{3^3} \cdot x^{0} \]

### Step 4: Simplify Further
Since \(x^0 = 1\):
\[ = \frac{1}{3^3} \cdot 1 = \frac{1}{27} \]

### Final Answer
The simplified expression is:
\[ \frac{1}{27} \]

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