Find the value of
`((243)^(0.13) xx (243)^(0.07))/((7)^(0.25) xx (49)^(0.075) xx (343)^(0.2))`

Answer

To solve the expression

\[ \frac{(243)^{0.13} \times (243)^{0.07}}{(7)^{0.25} \times (49)^{0.075} \times (343)^{0.2}} \]

we can follow these steps:

### Step 1: Simplify the Numerator
Since the bases in the numerator are the same (both are 243), we can add the exponents:

\[ (243)^{0.13} \times (243)^{0.07} = (243)^{0.13 + 0.07} = (243)^{0.20} \]

### Step 2: Simplify the Denominator
Now, let's simplify the denominator. We can express 49 and 343 in terms of their prime bases:

- \(49 = 7^2\)
- \(343 = 7^3\)

Now we can rewrite the denominator:

\[ (7)^{0.25} \times (49)^{0.075} \times (343)^{0.2} = (7)^{0.25} \times (7^2)^{0.075} \times (7^3)^{0.2} \]

Using the power of a power property, we can simplify further:

\[ (7)^{0.25} \times (7^{2 \times 0.075}) \times (7^{3 \times 0.2}) = (7)^{0.25} \times (7^{0.15}) \times (7^{0.6}) \]

### Step 3: Combine the Exponents in the Denominator
Now, we can add the exponents in the denominator since the bases are the same:

\[ (7)^{0.25 + 0.15 + 0.6} = (7)^{1.0} = 7 \]

### Step 4: Substitute Back into the Expression
Now we can substitute back into the original expression:

\[ \frac{(243)^{0.20}}{7} \]

### Step 5: Rewrite 243 in Terms of Base 3
Next, we can rewrite 243 as \(3^5\):

\[ 243 = 3^5 \Rightarrow (243)^{0.20} = (3^5)^{0.20} = 3^{5 \times 0.20} = 3^{1} = 3 \]

### Step 6: Final Expression
Now we can substitute this back into our expression:

\[ \frac{3}{7} \]

### Conclusion
Thus, the value of the given expression is

\[ \frac{3}{7} \]