Evaluate : `(27)^(-(1)/(3)).(27)^(-(1)/(3))[(27)^((1)/(3))-(27)^((2)/(3))]`

Options

A. `-2//3`
B. `1//3`
C. `-1//3`
D. `2//3`

Answer

To evaluate the expression \( (27)^{-\frac{1}{3}} \cdot (27)^{-\frac{1}{3}} \left[ (27)^{\frac{1}{3}} - (27)^{\frac{2}{3}} \right] \), we can follow these steps:

### Step 1: Rewrite 27 in terms of its prime factors
We know that \( 27 = 3^3 \). Therefore, we can rewrite the expression using this prime factorization.

### Step 2: Substitute 27 with \( 3^3 \)
The expression becomes:
\[ (3^3)^{-\frac{1}{3}} \cdot (3^3)^{-\frac{1}{3}} \left[ (3^3)^{\frac{1}{3}} - (3^3)^{\frac{2}{3}} \right] \]

### Step 3: Simplify the powers
Using the power of a power property \( (a^m)^n = a^{m \cdot n} \):
\[ (3^{3 \cdot -\frac{1}{3}}) \cdot (3^{3 \cdot -\frac{1}{3}}) \left[ (3^{3 \cdot \frac{1}{3}}) - (3^{3 \cdot \frac{2}{3}}) \right] \]
This simplifies to:
\[ (3^{-1}) \cdot (3^{-1}) \left[ (3^{1}) - (3^{2}) \right] \]

### Step 4: Combine the terms
Now we can simplify:
\[ 3^{-1} \cdot 3^{-1} = 3^{-2} \]
And the bracket simplifies to:
\[ 3 - 9 = -6 \]
So the expression now looks like:
\[ 3^{-2} \cdot (-6) \]

### Step 5: Simplify further
We know that \( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \). Therefore, we have:
\[ \frac{1}{9} \cdot (-6) = -\frac{6}{9} = -\frac{2}{3} \]

### Final Answer
Thus, the final answer is:
\[ -\frac{2}{3} \]