The value of expression `((1)/(3))^(3) xx ((-2)/(5))^(2) xx ((-3)/(2))^(3)` is

Options

A. `(1)/(50)`
B. `(-1)/(50)`
C. `1`
D. `0`

Answer

To solve the expression \(\left(\frac{1}{3}\right)^{3} \times \left(-\frac{2}{5}\right)^{2} \times \left(-\frac{3}{2}\right)^{3}\), we will break it down step by step.

### Step 1: Calculate each part of the expression

1. **Calculate \(\left(\frac{1}{3}\right)^{3}\)**:
\[ \left(\frac{1}{3}\right)^{3} = \frac{1^{3}}{3^{3}} = \frac{1}{27} \]

2. **Calculate \(\left(-\frac{2}{5}\right)^{2}\)**:
\[ \left(-\frac{2}{5}\right)^{2} = \frac{(-2)^{2}}{5^{2}} = \frac{4}{25} \]

3. **Calculate \(\left(-\frac{3}{2}\right)^{3}\)**:
\[ \left(-\frac{3}{2}\right)^{3} = \frac{(-3)^{3}}{2^{3}} = \frac{-27}{8} \]

### Step 2: Combine the results

Now, we combine all parts:
\[ \frac{1}{27} \times \frac{4}{25} \times \frac{-27}{8} \]

### Step 3: Multiply the fractions

We can multiply the numerators and the denominators:
\[ = \frac{1 \times 4 \times (-27)}{27 \times 25 \times 8} \]
\[ = \frac{-108}{27 \times 25 \times 8} \]

### Step 4: Simplify the expression

1. **Calculate the denominator**:
\[ 27 \times 25 = 675 \]
\[ 675 \times 8 = 5400 \]

2. **Now we have**:
\[ = \frac{-108}{5400} \]

3. **Simplify the fraction**:
To simplify \(\frac{-108}{5400}\), we can divide both the numerator and the denominator by 108:
\[ = \frac{-1}{50} \]

### Final Answer:
The value of the expression is \(\frac{-1}{50}\).

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