To solve the expression $2^a \cdot 8^{a+1}$, we can follow these steps:

1. Express 8 as a power of 2: $8 = 2^3$

   Therefore, $8^{a+1} = (2^3)^{a+1}$

2. Apply the power of a power property: $(2^3)^{a+1} = 2^{3(a+1)} = 2^{3a + 3}$

3. Rewrite the original expression using the above result: $2^a \cdot 8^{a+1} = 2^a \cdot 2^{3a + 3}$

4. Combine the exponents using the product of powers property: $2^a \cdot 2^{3a + 3} = 2^{a + 3a + 3} = 2^{4a + 3}$

So, the simplified form of the given expression $2^a \cdot 8^{a+1}$ is: $2^{4a + 3}$

Would you like any further details or have any questions about this solution?

Here are 8 related questions to expand on this information:

1. What is the general rule for combining exponents when multiplying powers of the same base?
2. How would the expression change if the base 2 were replaced with a different base, such as 3?
3. Can you show another example where you simplify an expression involving exponents and multiplication?
4. How does the power of a power property work in general?
5. What if the exponent of the second term was different, such as $8^{2a+1}$?
6. How can you simplify expressions involving roots and exponents?
7. How would you handle the expression if it involved division instead of multiplication?
8. What is the impact on the expression if $a$ is a negative number?

Tip: Always remember to express all terms with the same base before applying the properties of exponents to simplify expressions.