Simplify and express each of the following as a rational number :
` ((7)/(11))^(6) +(( 7)/(11))^(3)`

Answer

To simplify the expression \(\left(\frac{7}{11}\right)^{6} + \left(\frac{7}{11}\right)^{3}\) and express it as a rational number, we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the expression**:
\[ \left(\frac{7}{11}\right)^{6} + \left(\frac{7}{11}\right)^{3} \]

2. **Use the property of exponents**:
We can express the terms separately:
\[ = \frac{7^{6}}{11^{6}} + \frac{7^{3}}{11^{3}} \]

3. **Find a common denominator**:
The common denominator for both fractions is \(11^{6}\). We need to express the second term with this common denominator:
\[ = \frac{7^{6}}{11^{6}} + \frac{7^{3} \cdot 11^{3}}{11^{6}} \]

4. **Combine the fractions**:
Now that both fractions have the same denominator, we can add the numerators:
\[ = \frac{7^{6} + 7^{3} \cdot 11^{3}}{11^{6}} \]

5. **Factor out \(7^{3}\) from the numerator**:
We can factor \(7^{3}\) out of the numerator:
\[ = \frac{7^{3}(7^{3} + 11^{3})}{11^{6}} \]

6. **Final expression**:
Thus, the simplified expression is:
\[ = \frac{7^{3}(7^{3} + 11^{3})}{11^{6}} \]