Simplify and express the result as a rational number in each case :
` ( (-3)/(2))^(3) xx ( (-3)/(2))^(4)`

Answer

To simplify the expression \( \left( \frac{-3}{2} \right)^3 \times \left( \frac{-3}{2} \right)^4 \), we can follow these steps:

### Step 1: Identify the bases and exponents
The bases in both terms are the same, which is \( \frac{-3}{2} \). The exponents are 3 and 4.

### Step 2: Apply the Law of Exponents
According to the law of exponents, when multiplying two powers with the same base, we can add the exponents:
\[ x^a \times x^b = x^{a+b} \]
In our case:
\[ \left( \frac{-3}{2} \right)^3 \times \left( \frac{-3}{2} \right)^4 = \left( \frac{-3}{2} \right)^{3+4} \]

### Step 3: Simplify the exponent
Now we can add the exponents:
\[ 3 + 4 = 7 \]
So we have:
\[ \left( \frac{-3}{2} \right)^{7} \]

### Step 4: Calculate the power
Now we need to calculate \( \left( \frac{-3}{2} \right)^{7} \):
\[ \left( \frac{-3}{2} \right)^{7} = \frac{(-3)^7}{2^7} \]

### Step 5: Calculate \( (-3)^7 \) and \( 2^7 \)
Calculating \( (-3)^7 \):
\[ (-3)^7 = -2187 \]
Calculating \( 2^7 \):
\[ 2^7 = 128 \]

### Step 6: Combine the results
Now we can combine the results:
\[ \left( \frac{-3}{2} \right)^{7} = \frac{-2187}{128} \]

### Final Result
Thus, the simplified expression is:
\[ \frac{-2187}{128} \]

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