Simplify and express each of the following as a reational number :
` ((-5)/(8))^(11) + ( (-5)/(8))^(8) `

Answer

To simplify the expression \(\left(-\frac{5}{8}\right)^{11} + \left(-\frac{5}{8}\right)^{8}\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \left(-\frac{5}{8}\right)^{11} + \left(-\frac{5}{8}\right)^{8} \]

### Step 2: Apply the exponent rule
Using the property of exponents \(\left(\frac{A}{B}\right)^{M} = \frac{A^{M}}{B^{M}}\), we can rewrite each term:
\[ \frac{(-5)^{11}}{8^{11}} + \frac{(-5)^{8}}{8^{8}} \]

### Step 3: Make the denominators the same
To add the fractions, we need a common denominator. The first term has a denominator of \(8^{11}\) and the second term has \(8^{8}\). We can express \(8^{8}\) in terms of \(8^{11}\) by multiplying the second term by \(\frac{8^{3}}{8^{3}}\):
\[ \frac{(-5)^{11}}{8^{11}} + \frac{(-5)^{8} \cdot 8^{3}}{8^{11}} \]

### Step 4: Combine the fractions
Now that both fractions have the same denominator, we can combine them:
\[ \frac{(-5)^{11} + (-5)^{8} \cdot 8^{3}}{8^{11}} \]

### Step 5: Factor out the common term
Notice that \((-5)^{8}\) is a common factor in the numerator:
\[ \frac{(-5)^{8} \left((-5)^{3} + 8^{3}\right)}{8^{11}} \]

### Step 6: Simplify the expression inside the brackets
Now we can simplify \((-5)^{3}\) and \(8^{3}\):
\[ (-5)^{3} = -125 \quad \text{and} \quad 8^{3} = 512 \]
Thus, we have:
\[ (-5)^{3} + 8^{3} = -125 + 512 = 387 \]

### Step 7: Substitute back into the expression
Now we substitute this back into our expression:
\[ \frac{(-5)^{8} \cdot 387}{8^{11}} \]

### Step 8: Final expression
The final expression is:
\[ \frac{(-5)^{8} \cdot 387}{8^{11}} \]
Since \((-5)^{8}\) is positive (as it is raised to an even power), we can express it as:
\[ \frac{5^{8} \cdot 387}{8^{11}} \]

### Conclusion
This is the simplified form expressed as a rational number.

---