Simplify : `((243)^(3)xx(243)^7)/(7^(5)xx(49)^(5)xx(343)^2)`

Answer

To simplify the expression \(\frac{(243)^3 \times (243)^7}{7^5 \times (49)^5 \times (343)^2}\), we can follow these steps:

### Step 1: Simplify the Numerator
First, we can combine the powers of \(243\) in the numerator:
\[ (243)^3 \times (243)^7 = (243)^{3+7} = (243)^{10} \]

### Step 2: Factor \(243\)
Next, we need to express \(243\) in terms of its prime factors. We know that:
\[ 243 = 3^5 \]
Thus, we can rewrite the numerator:
\[ (243)^{10} = (3^5)^{10} = 3^{5 \times 10} = 3^{50} \]

### Step 3: Simplify the Denominator
Now, let's simplify the denominator:
\[ 7^5 \times (49)^5 \times (343)^2 \]
We can factor \(49\) and \(343\) as follows:
\[ 49 = 7^2 \quad \text{and} \quad 343 = 7^3 \]
Now substituting these values in:
\[ (49)^5 = (7^2)^5 = 7^{2 \times 5} = 7^{10} \]
\[ (343)^2 = (7^3)^2 = 7^{3 \times 2} = 7^6 \]
Combining these in the denominator:
\[ 7^5 \times 7^{10} \times 7^6 = 7^{5 + 10 + 6} = 7^{21} \]

### Step 4: Combine the Results
Now we can combine the results from the numerator and the denominator:
\[ \frac{3^{50}}{7^{21}} \]

### Final Answer
Thus, the simplified form of the expression is:
\[ \frac{3^{50}}{7^{21}} \]

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