Simplify and express each of the following as a rational number :
` (3^(5) xx 25 xx 10 ^(5))/( 5^(7) xx 6^(5))`

Answer

To simplify the expression \( \frac{3^{5} \times 25 \times 10^{5}}{5^{7} \times 6^{5}} \) and express it as a rational number, we will follow these steps:

### Step 1: Factorize each number into its prime factors.
- \( 3^{5} \) remains as it is since 3 is a prime number.
- \( 25 = 5 \times 5 = 5^{2} \).
- \( 10 = 2 \times 5 \), so \( 10^{5} = (2 \times 5)^{5} = 2^{5} \times 5^{5} \).
- \( 5^{7} \) remains as it is since 5 is a prime number.
- \( 6 = 2 \times 3 \), so \( 6^{5} = (2 \times 3)^{5} = 2^{5} \times 3^{5} \).

### Step 2: Substitute the prime factorizations into the expression.
Now, substituting these factorizations into the original expression, we get:
\[ \frac{3^{5} \times 5^{2} \times (2^{5} \times 5^{5})}{5^{7} \times (2^{5} \times 3^{5})} \]

### Step 3: Simplify the expression.
This simplifies to:
\[ \frac{3^{5} \times 5^{2} \times 2^{5} \times 5^{5}}{5^{7} \times 2^{5} \times 3^{5}} \]

### Step 4: Cancel out common factors.
- The \( 3^{5} \) in the numerator and denominator cancels out.
- The \( 2^{5} \) in the numerator and denominator cancels out.
- Now we have:
\[ \frac{5^{2} \times 5^{5}}{5^{7}} \]

### Step 5: Apply the law of exponents.
Using the law of exponents \( a^{m} \times a^{n} = a^{m+n} \):
\[ 5^{2} \times 5^{5} = 5^{2 + 5} = 5^{7} \]
Thus, we can rewrite the expression as:
\[ \frac{5^{7}}{5^{7}} \]

### Step 6: Simplify the fraction.
Since \( \frac{5^{7}}{5^{7}} = 1 \), the final result is:
\[ 1 \]

### Final Answer:
The simplified expression is \( 1 \).