`((27)/(343))^(2/3)xx((343)/(729))^(2/3)div((2401)/(81))^(3/4)`= _____

Options

A. `1//3xx(7//3)^8`
B. `1//9xx(7//3)^7`
C. `1//9xx(7//3)^6`
D. `(7//3)^(6)`

Explanation

To solve the expression \(\left(\frac{27}{343}\right)^{\frac{2}{3}} \times \left(\frac{343}{729}\right)^{\frac{2}{3}} \div \left(\frac{2401}{81}\right)^{\frac{3}{4}}\), we will simplify it step by step.

### Step 1: Rewrite the numbers in terms of their bases
We know that:
- \(27 = 3^3\)
- \(343 = 7^3\)
- \(729 = 3^6\)
- \(2401 = 7^4\)
- \(81 = 3^4\)

So we can rewrite the expression as:
\[ \left(\frac{3^3}{7^3}\right)^{\frac{2}{3}} \times \left(\frac{7^3}{3^6}\right)^{\frac{2}{3}} \div \left(\frac{7^4}{3^4}\right)^{\frac{3}{4}} \]

### Step 2: Apply the power of a quotient rule
Using the property \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\), we can simplify each term:
\[ \frac{(3^3)^{\frac{2}{3}}}{(7^3)^{\frac{2}{3}}} \times \frac{(7^3)^{\frac{2}{3}}}{(3^6)^{\frac{2}{3}}} \div \frac{(7^4)^{\frac{3}{4}}}{(3^4)^{\frac{3}{4}}} \]

### Step 3: Simplify the powers
Calculating the powers:
\[ \frac{3^{3 \cdot \frac{2}{3}}}{7^{3 \cdot \frac{2}{3}}} \times \frac{7^{3 \cdot \frac{2}{3}}}{3^{6 \cdot \frac{2}{3}}} \div \frac{7^{4 \cdot \frac{3}{4}}}{3^{4 \cdot \frac{3}{4}}} \]
This simplifies to:
\[ \frac{3^2}{7^2} \times \frac{7^2}{3^4} \div \frac{7^3}{3^3} \]

### Step 4: Combine the fractions
Now we can combine the fractions:
\[ \frac{3^2 \cdot 7^2}{7^2 \cdot 3^4} \div \frac{7^3}{3^3} \]
This simplifies to:
\[ \frac{3^2}{3^4} = \frac{1}{3^2} = \frac{1}{9} \]
And for the division:
\[ \div \frac{7^3}{3^3} = \times \frac{3^3}{7^3} \]
So we have:
\[ \frac{1}{9} \times \frac{3^3}{7^3} = \frac{3^3}{9 \cdot 7^3} \]

### Step 5: Simplify further
Since \(9 = 3^2\), we can write:
\[ \frac{3^3}{3^2 \cdot 7^3} = \frac{3^{3-2}}{7^3} = \frac{3^1}{7^3} = \frac{3}{343} \]

### Final Answer
Thus, the final answer is:
\[ \frac{3}{343} \]