Express each of the following as a rational number of the form `(p)/(q)`.
`((-2)/(7))^(-4) xx ((-7)/(5))^(3)`

Answer

To express the given expression \(\left(-\frac{2}{7}\right)^{-4} \times \left(-\frac{7}{5}\right)^{3}\) as a rational number of the form \(\frac{p}{q}\), we will follow these steps:

### Step 1: Apply the property of negative exponents
The property states that \(a^{-m} = \frac{1}{a^m}\). Therefore, we can rewrite \(\left(-\frac{2}{7}\right)^{-4}\) as:
\[ \left(-\frac{2}{7}\right)^{-4} = \frac{1}{\left(-\frac{2}{7}\right)^{4}} \]

### Step 2: Calculate \(\left(-\frac{2}{7}\right)^{4}\)
Now we need to calculate \(\left(-\frac{2}{7}\right)^{4}\):
\[ \left(-\frac{2}{7}\right)^{4} = \frac{(-2)^{4}}{7^{4}} = \frac{16}{2401} \]
Thus,
\[ \left(-\frac{2}{7}\right)^{-4} = \frac{1}{\frac{16}{2401}} = \frac{2401}{16} \]

### Step 3: Calculate \(\left(-\frac{7}{5}\right)^{3}\)
Next, we calculate \(\left(-\frac{7}{5}\right)^{3}\):
\[ \left(-\frac{7}{5}\right)^{3} = \frac{(-7)^{3}}{5^{3}} = \frac{-343}{125} \]

### Step 4: Multiply the two results
Now we multiply the two results:
\[ \frac{2401}{16} \times \frac{-343}{125} = \frac{2401 \times -343}{16 \times 125} \]

### Step 5: Calculate the numerator and denominator
Calculating the numerator:
\[ 2401 \times -343 = -823543 \]
Calculating the denominator:
\[ 16 \times 125 = 2000 \]

### Step 6: Write the final result
Thus, we have:
\[ \frac{-823543}{2000} \]

### Final Answer
The expression \(\left(-\frac{2}{7}\right)^{-4} \times \left(-\frac{7}{5}\right)^{3}\) can be expressed as:
\[ \frac{-823543}{2000} \]