Simplify and express each of the following as a rational number :
` {((-3)/( 2))^(3)} ^(2) `

Answer

To simplify the expression \(\left(\frac{-3}{2}\right)^{3}\) raised to the power of \(2\), we can follow these steps:

### Step 1: Apply the Law of Exponents
According to the law of exponents, \((a^m)^n = a^{m \cdot n}\). Here, we can apply this law to our expression:

\[ \left(\frac{-3}{2}\right)^{3 \cdot 2} \]

### Step 2: Calculate the Exponents
Now, we multiply the exponents:

\[ 3 \cdot 2 = 6 \]

So, we can rewrite the expression as:

\[ \left(\frac{-3}{2}\right)^{6} \]

### Step 3: Calculate the Power
Next, we need to calculate \(\left(\frac{-3}{2}\right)^{6}\). This means we raise both the numerator and the denominator to the power of \(6\):

\[ \frac{(-3)^{6}}{(2)^{6}} \]

### Step 4: Calculate \((-3)^{6}\) and \(2^{6}\)
Now we calculate \((-3)^{6}\) and \(2^{6}\):

\[ (-3)^{6} = 729 \quad \text{(since the negative sign disappears when raised to an even power)} \]
\[ 2^{6} = 64 \]

### Step 5: Form the Final Expression
Now we can write the expression as:

\[ \frac{729}{64} \]

### Step 6: Conclusion
Thus, the simplified expression of \(\left(\frac{-3}{2}\right)^{3}\) raised to the power of \(2\) is:

\[ \frac{729}{64} \]