Simplify and express each of the following as a rational number :
` ((-7)/(11)) ^(17) + ( (-7)/( 11) ) ^(19)`

Answer

To simplify the expression \(\left(-\frac{7}{11}\right)^{17} + \left(-\frac{7}{11}\right)^{19}\) and express it as a rational number, we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \left(-\frac{7}{11}\right)^{17} + \left(-\frac{7}{11}\right)^{19} \]

### Step 2: Apply the Exponent Rule
Using the exponent rule \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\), we can rewrite each term:
\[ \frac{(-7)^{17}}{11^{17}} + \frac{(-7)^{19}}{11^{19}} \]

### Step 3: Find a Common Denominator
The common denominator for both fractions is \(11^{19}\). We need to express the first term with this denominator:
\[ \frac{(-7)^{17} \cdot 11^2}{11^{19}} + \frac{(-7)^{19}}{11^{19}} \]
Here, we multiplied the numerator of the first term by \(11^2\) (which is \(121\)) to match the denominator.

### Step 4: Combine the Fractions
Now that both fractions have the same denominator, we can combine them:
\[ \frac{(-7)^{17} \cdot 11^2 + (-7)^{19}}{11^{19}} \]

### Step 5: Factor Out Common Terms
Notice that \((-7)^{17}\) is common in the numerator:
\[ \frac{(-7)^{17} \left(11^2 + (-7)^2\right)}{11^{19}} \]

### Step 6: Simplify the Expression Inside the Bracket
Now, we simplify the expression inside the brackets:
\[ 11^2 = 121 \quad \text{and} \quad (-7)^2 = 49 \]
Thus,
\[ 11^2 + (-7)^2 = 121 + 49 = 170 \]

### Step 7: Substitute Back
Now substitute back into the expression:
\[ \frac{(-7)^{17} \cdot 170}{11^{19}} \]

### Step 8: Final Expression
The final expression is:
\[ \frac{(-7)^{17} \cdot 170}{11^{19}} \]

### Conclusion
This is a rational number since both the numerator and denominator are integers.