Simplify and express each of the following as a rational number :
` ((-3)^(11))/((-3)^(5))`

Answer

To simplify the expression \(\frac{(-3)^{11}}{(-3)^{5}}\), we can follow these steps:

### Step 1: Identify the base and the exponents
The base here is \(-3\) and the exponents are \(11\) for the numerator and \(5\) for the denominator.

### Step 2: Apply the exponent rule for division
According to the exponent rule, \(\frac{a^m}{a^n} = a^{m-n}\). Here, we can apply this rule since the bases are the same.

\[ \frac{(-3)^{11}}{(-3)^{5}} = (-3)^{11-5} \]

### Step 3: Simplify the exponent
Now, we simplify the exponent:

\[ (-3)^{11-5} = (-3)^{6} \]

### Step 4: Calculate \((-3)^6\)
Next, we calculate \((-3)^6\). Since the exponent is even, the result will be positive:

\[ (-3)^6 = 729 \]

### Step 5: Write the final answer as a rational number
Since \(729\) can be expressed as a fraction, we write it as:

\[ \frac{729}{1} \]

Thus, the simplified expression is:

\[ \frac{729}{1} \]

### Final Answer:
\(\frac{729}{1}\) is a rational number.

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