Simplify and express each of the following as a rational number :
` (10^(2) xx 15^(3))/( 2^(2) xx 3xx 5^(5) xx 6^(4))`

Answer

To simplify the expression \( \frac{10^2 \times 15^3}{2^2 \times 3 \times 5^5 \times 6^4} \), we will follow these steps:

### Step 1: Factor the composite numbers
First, we need to express the composite numbers (10, 15, and 6) in terms of their prime factors:
- \( 10 = 2 \times 5 \)
- \( 15 = 3 \times 5 \)
- \( 6 = 2 \times 3 \)

### Step 2: Rewrite the expression
Now, we can rewrite the expression using these prime factors:
\[ 10^2 = (2 \times 5)^2 = 2^2 \times 5^2 \]
\[ 15^3 = (3 \times 5)^3 = 3^3 \times 5^3 \]
\[ 6^4 = (2 \times 3)^4 = 2^4 \times 3^4 \]
Thus, the expression becomes:
\[ \frac{(2^2 \times 5^2) \times (3^3 \times 5^3)}{2^2 \times 3 \times 5^5 \times (2^4 \times 3^4)} \]

### Step 3: Combine the numerator
Now, let's combine the terms in the numerator:
\[ = \frac{2^2 \times 5^2 \times 3^3 \times 5^3}{2^2 \times 3 \times 5^5 \times 2^4 \times 3^4} \]
\[ = \frac{2^2 \times 3^3 \times 5^{2+3}}{2^{2+4} \times 3^{1+4} \times 5^5} \]
\[ = \frac{2^2 \times 3^3 \times 5^5}{2^6 \times 3^5 \times 5^5} \]

### Step 4: Simplify the expression
Now we can simplify the expression by canceling out the common terms in the numerator and denominator:
\[ = \frac{2^2}{2^6} \times \frac{3^3}{3^5} \times \frac{5^5}{5^5} \]
Using the laws of exponents:
\[ = 2^{2-6} \times 3^{3-5} \times 5^{5-5} \]
\[ = 2^{-4} \times 3^{-2} \times 5^0 \]

### Step 5: Convert negative exponents to positive
Using the property \( x^{-a} = \frac{1}{x^a} \):
\[ = \frac{1}{2^4} \times \frac{1}{3^2} \times 1 \]
\[ = \frac{1}{2^4 \times 3^2} \]

### Step 6: Calculate the values
Now we calculate \( 2^4 \) and \( 3^2 \):
\[ 2^4 = 16 \quad \text{and} \quad 3^2 = 9 \]
Thus, we have:
\[ = \frac{1}{16 \times 9} = \frac{1}{144} \]

### Final Answer
The simplified expression is:
\[ \frac{1}{144} \]

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