Evaluate `((-2)/(7)) ^(-4)xx ((-5)/(7)) ^(2).`

Answer

To evaluate the expression \(\left(-\frac{2}{7}\right)^{-4} \times \left(-\frac{5}{7}\right)^{2}\), we can follow these steps:

### Step 1: Apply the negative exponent rule
The first part of the expression has a negative exponent. According to the rule of exponents, \(a^{-m} = \frac{1}{a^{m}}\). Therefore, we can rewrite \(\left(-\frac{2}{7}\right)^{-4}\) as:

\[ \left(-\frac{2}{7}\right)^{-4} = \frac{1}{\left(-\frac{2}{7}\right)^{4}} = \frac{7^{4}}{(-2)^{4}} \]

### Step 2: Calculate \((-2)^{4}\) and \(7^{4}\)
Now we calculate \((-2)^{4}\) and \(7^{4}\):

\[ (-2)^{4} = 16 \quad \text{(since multiplying four negative twos gives a positive result)} \]
\[ 7^{4} = 7 \times 7 \times 7 \times 7 = 2401 \]

So we have:

\[ \left(-\frac{2}{7}\right)^{-4} = \frac{2401}{16} \]

### Step 3: Evaluate \(\left(-\frac{5}{7}\right)^{2}\)
Next, we evaluate the second part of the expression \(\left(-\frac{5}{7}\right)^{2}\):

\[ \left(-\frac{5}{7}\right)^{2} = \frac{(-5)^{2}}{7^{2}} = \frac{25}{49} \]

### Step 4: Combine the results
Now we can combine the two parts:

\[ \left(-\frac{2}{7}\right)^{-4} \times \left(-\frac{5}{7}\right)^{2} = \frac{2401}{16} \times \frac{25}{49} \]

### Step 5: Multiply the fractions
To multiply the fractions, we multiply the numerators and the denominators:

\[ \frac{2401 \times 25}{16 \times 49} \]

Calculating the numerator:

\[ 2401 \times 25 = 60025 \]

Calculating the denominator:

\[ 16 \times 49 = 784 \]

### Step 6: Final result
Thus, we have:

\[ \frac{60025}{784} \]

This fraction cannot be simplified further, so the final answer is:

\[ \frac{60025}{784} \]