Simplify and express each of the following as a rationla number .
` ((1)/(2))^(3) xx (( 3)/(2))^(2) xx 2^(3)`

Answer

To simplify the expression \(\left(\frac{1}{2}\right)^{3} \times \left(\frac{3}{2}\right)^{2} \times 2^{3}\), we will follow the laws of exponents step by step.

### Step-by-Step Solution:

1. **Rewrite the expression**:
\[ \left(\frac{1}{2}\right)^{3} \times \left(\frac{3}{2}\right)^{2} \times 2^{3} \]

2. **Apply the exponent rule for fractions**:
Using the property \(\left(\frac{x}{y}\right)^{a} = \frac{x^{a}}{y^{a}}\), we can rewrite \(\left(\frac{1}{2}\right)^{3}\) and \(\left(\frac{3}{2}\right)^{2}\):
\[ = \frac{1^{3}}{2^{3}} \times \frac{3^{2}}{2^{2}} \times 2^{3} \]

3. **Calculate the powers**:
\[ = \frac{1}{2^{3}} \times \frac{9}{2^{2}} \times 2^{3} \]

4. **Combine the fractions**:
Now we can combine the fractions:
\[ = \frac{1 \times 9}{2^{3} \times 2^{2}} \times 2^{3} \]

5. **Use the exponent rule for multiplication**:
According to the law \(a^{m} \times a^{n} = a^{m+n}\):
\[ = \frac{9}{2^{3+2}} \times 2^{3} = \frac{9}{2^{5}} \times 2^{3} \]

6. **Simplify the expression**:
\[ = \frac{9 \times 2^{3}}{2^{5}} = \frac{9 \times 2^{3}}{2^{3} \times 2^{2}} = \frac{9}{2^{2}} = \frac{9}{4} \]

7. **Final answer**:
The simplified expression is:
\[ \frac{9}{4} \]