Let's analyze the two given quantities to compare them.

Quantity A:

$\frac{8^{8-4}}{8^3} = \frac{8^4}{8^3}$ Using the properties of exponents, we can simplify this: $\frac{8^4}{8^3} = 8^{4-3} = 8^1 = 8$ So, Quantity A = 8.

Quantity B:

$\frac{1}{2^{15}}$ This is just the reciprocal of $2^{15}$, which is a very small number: $\frac{1}{2^{15}} \approx 0.00003$

Comparison:

Clearly, Quantity A (which is 8) is much larger than Quantity B (which is approximately 0.00003).

Thus, the correct answer is: $\boxed{\text{Quantity A is greater.}}$

Would you like further details or have any questions? Here are 5 related questions you could explore:

1. What is the general rule for simplifying expressions with exponents?
2. How does the size of exponents affect the comparison of large or small numbers?
3. Can you explain why $2^{15}$ grows so quickly?
4. How can you compare powers of different bases, like powers of 2 and powers of 3?
5. How would you handle expressions involving fractional exponents?

Tip: Always remember that when dividing like bases, you subtract the exponents!