Simplify and express each of the following as a rational number :
` ((3)/(4))^(3) xx ((-1)/( 2))^(5) xx 2^(3)`

Answer

To simplify the expression \( \left(\frac{3}{4}\right)^{3} \times \left(-\frac{1}{2}\right)^{5} \times 2^{3} \) and express it as a rational number, we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \left(\frac{3}{4}\right)^{3} \times \left(-\frac{1}{2}\right)^{5} \times 2^{3} \]
We can rewrite \( 4 \) as \( 2^{2} \):
\[ \left(\frac{3}{2^{2}}\right)^{3} \times \left(-\frac{1}{2}\right)^{5} \times 2^{3} \]

### Step 2: Apply the exponent to the fraction
Using the property \( \left(\frac{x}{y}\right)^{a} = \frac{x^{a}}{y^{a}} \):
\[ \frac{3^{3}}{(2^{2})^{3}} \times \left(-1\right)^{5} \times \frac{1}{2^{5}} \times 2^{3} \]
This simplifies to:
\[ \frac{3^{3}}{2^{6}} \times (-1) \times \frac{1}{2^{5}} \times 2^{3} \]

### Step 3: Simplify the expression
Now we can combine the terms:
\[ \frac{3^{3} \times (-1)}{2^{6} \times 2^{5}} \times 2^{3} \]
Using the property \( x^{a} \times x^{b} = x^{a+b} \):
\[ \frac{3^{3} \times (-1)}{2^{6 + 5 - 3}} = \frac{3^{3} \times (-1)}{2^{8}} \]

### Step 4: Calculate the powers
Now we calculate \( 3^{3} \):
\[ 3^{3} = 27 \]
Thus, we have:
\[ \frac{-27}{2^{8}} = \frac{-27}{256} \]

### Final Answer
The final simplified expression is:
\[ \frac{-27}{256} \]

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