Simplify and express each of the following as a rational number :
` (7^(2))/( 7^(5)) `

Answer

To simplify the expression \( \frac{7^2}{7^5} \) and express it as a rational number, follow these steps:

### Step-by-step Solution:

1. **Identify the expression**: We have \( \frac{7^2}{7^5} \).

2. **Use the property of exponents**: When dividing two powers with the same base, we subtract the exponents. This can be expressed as:
\[ \frac{x^a}{x^b} = x^{a-b} \]
Here, \( a = 2 \) and \( b = 5 \). So, we can rewrite the expression as:
\[ 7^{2-5} \]

3. **Perform the subtraction**: Calculate \( 2 - 5 \):
\[ 2 - 5 = -3 \]
Therefore, we have:
\[ 7^{-3} \]

4. **Convert to positive exponent**: To express \( 7^{-3} \) as a positive exponent, we can use the property that \( x^{-a} = \frac{1}{x^a} \):
\[ 7^{-3} = \frac{1}{7^3} \]

5. **Calculate \( 7^3 \)**: Now we need to calculate \( 7^3 \):
\[ 7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343 \]
Thus, we have:
\[ \frac{1}{7^3} = \frac{1}{343} \]

6. **Final answer**: The expression \( \frac{7^2}{7^5} \) simplifies to:
\[ \frac{1}{343} \]
This is a rational number, expressed in the form \( \frac{V}{Q} \), where \( Q \neq 0 \).