Simplify :
`((3)/(7))^(-2) xx ((7)/(6))^(-3)`

Answer

To simplify the expression \(\left(\frac{3}{7}\right)^{-2} \times \left(\frac{7}{6}\right)^{-3}\), we can follow these steps:

### Step 1: Apply the property of negative exponents
Recall that \(a^{-b} = \frac{1}{a^b}\). Therefore, we can rewrite the expression as:
\[ \left(\frac{3}{7}\right)^{-2} = \frac{1}{\left(\frac{3}{7}\right)^{2}} \quad \text{and} \quad \left(\frac{7}{6}\right)^{-3} = \frac{1}{\left(\frac{7}{6}\right)^{3}} \]
So, we have:
\[ \left(\frac{3}{7}\right)^{-2} \times \left(\frac{7}{6}\right)^{-3} = \frac{1}{\left(\frac{3}{7}\right)^{2}} \times \frac{1}{\left(\frac{7}{6}\right)^{3}} = \frac{1}{\left(\frac{3}{7}\right)^{2} \times \left(\frac{7}{6}\right)^{3}} \]

### Step 2: Calculate the powers
Now, we compute the powers:
\[ \left(\frac{3}{7}\right)^{2} = \frac{3^2}{7^2} = \frac{9}{49} \]
\[ \left(\frac{7}{6}\right)^{3} = \frac{7^3}{6^3} = \frac{343}{216} \]

### Step 3: Substitute back into the expression
Now substituting these values back, we have:
\[ \frac{1}{\left(\frac{3}{7}\right)^{2} \times \left(\frac{7}{6}\right)^{3}} = \frac{1}{\frac{9}{49} \times \frac{343}{216}} \]

### Step 4: Multiply the fractions
To multiply the fractions, we multiply the numerators and the denominators:
\[ \frac{9}{49} \times \frac{343}{216} = \frac{9 \times 343}{49 \times 216} \]

### Step 5: Simplify the expression
Now we simplify:
\[ \frac{9 \times 343}{49 \times 216} = \frac{3087}{10584} \]
Next, we can simplify this fraction. Notice that \(49 = 7^2\) and \(343 = 7^3\), so:
\[ \frac{9 \times 7^3}{7^2 \times 216} = \frac{9 \times 7}{216} = \frac{63}{216} \]
Now we can simplify \(\frac{63}{216}\) by dividing both the numerator and denominator by 9:
\[ \frac{63 \div 9}{216 \div 9} = \frac{7}{24} \]

### Step 6: Final Result
Thus, the simplified expression is:
\[ \frac{24}{7} \]