Let's go through the problems one by one.

Part A:

Expression: $2^3 \times 2^8$

1. What is the common base?
   The common base is $2$, since both terms share the base $2$.

2. What is the sum of the exponents?
   We add the exponents using the Product of Powers Property:
   $3 + 8 = 11$.

3. What power is equivalent to the multiplication expression?
   The equivalent expression is:
   $2^{3+8} = 2^{11}$

Part B:

Expression: $5^9 \div 5^6$

4. What is the common base?
   The common base is $5$.

5. What is the difference of the exponents?
   We subtract the exponents using the Quotient of Powers Property:
   $9 - 6 = 3$.

6. What power is equivalent to the division expression?
   The equivalent expression is:
   $5^{9-6} = 5^3$

Would you like a more detailed explanation for any of these steps? Here are 5 related questions to expand on this concept:

1. What is the base in the expression $10^5 \times 10^3$?
2. If you have $3^4 \times 3^7$, what would the equivalent power be?
3. How would you simplify $4^8 \div 4^5$?
4. Can you apply the product of powers property to $7^2 \times 2^3$?
5. How does the quotient of powers property work when dividing powers of the same base with equal exponents?

Tip: Always ensure that the base of the numbers is the same before applying the product or quotient of powers properties.