Simplify and express each of the following as a rational number :
` ((-5) ^(21))/( (-5)^(25))`

Answer

To simplify the expression \(\frac{(-5)^{21}}{(-5)^{25}}\), we can follow these steps:

### Step 1: Apply the Quotient Rule of Exponents
Using the rule \(\frac{x^a}{x^b} = x^{a-b}\), we can simplify the expression:
\[ \frac{(-5)^{21}}{(-5)^{25}} = (-5)^{21 - 25} \]

### Step 2: Subtract the Exponents
Now, we calculate \(21 - 25\):
\[ 21 - 25 = -4 \]
So, we have:
\[ (-5)^{-4} \]

### Step 3: Convert to Positive Exponent
Using the rule \(x^{-a} = \frac{1}{x^a}\), we can rewrite the expression:
\[ (-5)^{-4} = \frac{1}{(-5)^4} \]

### Step 4: Calculate \((-5)^4\)
Now, we need to calculate \((-5)^4\):
\[ (-5)^4 = (-5) \times (-5) \times (-5) \times (-5) \]
Calculating this step by step:
- \((-5) \times (-5) = 25\)
- \(25 \times (-5) = -125\)
- \(-125 \times (-5) = 625\)

Thus, \((-5)^4 = 625\).

### Step 5: Substitute Back
Now we substitute back into our expression:
\[ \frac{1}{(-5)^4} = \frac{1}{625} \]

### Final Answer
The simplified expression is:
\[ \frac{1}{625} \]