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

Options

A. `(3)/(7)`
B. `(7)/(3)`
C. `1(3)/(7)`
D. `2(2)/(7)`

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 break it down step by step.

### Step 1: Simplify the Numerator
The numerator consists of two terms with the same base, \(243\). We can use the property of exponents that states \(a^m \times a^n = a^{m+n}\).

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

### Step 2: Rewrite the Bases in the Denominator
Next, we will rewrite the bases in the denominator in terms of powers of \(7\):
- \(49\) can be expressed as \(7^2\).
- \(343\) can be expressed as \(7^3\).

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 property of exponents \(a^m \times a^n = a^{m+n}\):

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

### Step 3: Combine the Exponents in the Denominator
Now we can combine the exponents in the denominator:

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

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

\[ \frac{(243)^{0.20}}{(7)^{1.0}} = \frac{(243)^{0.20}}{7} \]

### Step 5: Rewrite \(243\) in Terms of Base \(3\)
Next, we can express \(243\) as a power of \(3\):

\[ 243 = 3^5 \]

Thus,

\[ (243)^{0.20} = (3^5)^{0.20} = 3^{5 \times 0.20} = 3^{1.0} = 3 \]

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

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

### Conclusion
The final value of the expression is:

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