If `a^(7)=7,777 and (a^(6))/(b)=11`, what is the value of ab?

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations**:
- \( a^7 = 7777 \)
- \( \frac{a^6}{b} = 11 \)

2. **Find the value of \( a \)**:
- From the first equation, we can express \( a \) as:
\[ a = 7777^{\frac{1}{7}} \]

3. **Find the value of \( b \)**:
- Rearranging the second equation gives:
\[ b = \frac{a^6}{11} \]
- Now, we substitute \( a \) into this equation:
\[ b = \frac{(7777^{\frac{1}{7}})^6}{11} \]
- Simplifying \( (7777^{\frac{1}{7}})^6 \):
\[ b = \frac{7777^{\frac{6}{7}}}{11} \]

4. **Find the value of \( ab \)**:
- Now we calculate \( ab \):
\[ ab = a \cdot b = 7777^{\frac{1}{7}} \cdot \frac{7777^{\frac{6}{7}}}{11} \]
- This can be simplified as:
\[ ab = \frac{7777^{\frac{1}{7} + \frac{6}{7}}}{11} = \frac{7777^{\frac{7}{7}}}{11} = \frac{7777}{11} \]

5. **Final Calculation**:
- Now we compute \( \frac{7777}{11} \):
\[ 7777 \div 11 = 707 \]

Thus, the value of \( ab \) is \( 707 \).