Evaluate:
`((5)/(8))^(-7) xx ((8)/(5))^(-4)`

Answer

To evaluate the expression \(\left(\frac{5}{8}\right)^{-7} \times \left(\frac{8}{5}\right)^{-4}\), we will follow these steps:

### Step 1: Apply the property of negative exponents
We know that \(a^{-m} = \frac{1}{a^m}\). Therefore, we can rewrite the expression as:
\[ \left(\frac{5}{8}\right)^{-7} = \frac{1}{\left(\frac{5}{8}\right)^{7}} \quad \text{and} \quad \left(\frac{8}{5}\right)^{-4} = \frac{1}{\left(\frac{8}{5}\right)^{4}} \]
Thus, the expression becomes:
\[ \frac{1}{\left(\frac{5}{8}\right)^{7}} \times \frac{1}{\left(\frac{8}{5}\right)^{4}} = \frac{1}{\left(\frac{5}{8}\right)^{7} \times \left(\frac{8}{5}\right)^{4}} \]

### Step 2: Rewrite the fractions with positive exponents
Using the property \(\left(\frac{a}{b}\right)^{m} = \frac{a^{m}}{b^{m}}\), we can rewrite the terms:
\[ \left(\frac{5}{8}\right)^{7} = \frac{5^{7}}{8^{7}} \quad \text{and} \quad \left(\frac{8}{5}\right)^{4} = \frac{8^{4}}{5^{4}} \]
Thus, we have:
\[ \frac{1}{\frac{5^{7}}{8^{7}} \times \frac{8^{4}}{5^{4}}} \]

### Step 3: Combine the fractions
Now, we can combine the fractions:
\[ \frac{1}{\frac{5^{7} \times 8^{4}}{8^{7} \times 5^{4}}} = \frac{8^{7} \times 5^{4}}{5^{7} \times 8^{4}} \]

### Step 4: Simplify the expression
Using the property \(\frac{a^{m}}{a^{n}} = a^{m-n}\), we can simplify:
\[ \frac{8^{7}}{8^{4}} \times \frac{5^{4}}{5^{7}} = 8^{7-4} \times 5^{4-7} = 8^{3} \times 5^{-3} \]

### Step 5: Rewrite with positive exponents
This can be rewritten as:
\[ \frac{8^{3}}{5^{3}} \]

### Step 6: Calculate the values
Now, we calculate \(8^{3}\) and \(5^{3}\):
\[ 8^{3} = 512 \quad \text{and} \quad 5^{3} = 125 \]
Thus, we have:
\[ \frac{512}{125} \]

### Final Answer
The evaluated expression is:
\[ \frac{512}{125} \]