Simplify : `27^(-(1)/(3))(27^((1)/(3))-27^((2)/(3)))`

Answer

To simplify the expression \( 27^{-\frac{1}{3}}(27^{\frac{1}{3}} - 27^{\frac{2}{3}}) \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 27^{-\frac{1}{3}}(27^{\frac{1}{3}} - 27^{\frac{2}{3}}) \]

### Step 2: Distribute \( 27^{-\frac{1}{3}} \)
Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we distribute \( 27^{-\frac{1}{3}} \) to both terms inside the parentheses:
\[ 27^{-\frac{1}{3}} \cdot 27^{\frac{1}{3}} - 27^{-\frac{1}{3}} \cdot 27^{\frac{2}{3}} \]

### Step 3: Simplify each term
Now, we simplify each term using the property of exponents:
1. For the first term:
\[ 27^{-\frac{1}{3} + \frac{1}{3}} = 27^{0} = 1 \]
2. For the second term:
\[ 27^{-\frac{1}{3} + \frac{2}{3}} = 27^{\frac{1}{3}} \]

### Step 4: Combine the results
Now we can combine the results from the simplifications:
\[ 1 - 27^{\frac{1}{3}} \]

### Step 5: Calculate \( 27^{\frac{1}{3}} \)
The cube root of 27 is:
\[ 27^{\frac{1}{3}} = 3 \]

### Step 6: Final simplification
Now substituting back, we have:
\[ 1 - 3 = -2 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{-2} \]

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