We are given the expression:

$\frac{16^8}{16^6} \div 32^2$

Step 1: Simplify the first part of the expression

Using the exponent rule for division, $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{16^8}{16^6} = 16^{8-6} = 16^2$

Step 2: Simplify 32 as a power of 2

Since $16 = 2^4$ and $32 = 2^5$, we can rewrite the expression:

$16^2 = (2^4)^2 = 2^{8}$ $32^2 = (2^5)^2 = 2^{10}$

Step 3: Rewrite the expression

Now, the expression becomes:

$\frac{2^8}{2^{10}}$

Step 4: Apply the division rule

Using the exponent rule $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{2^8}{2^{10}} = 2^{8-10} = 2^{-2}$

Step 5: Final answer

Since $2^{-2} = \frac{1}{2^2} = \frac{1}{4}$, the final answer is:

$\frac{1}{4}$

Summary

The simplified result of the expression $\frac{16^8}{16^6} \div 32^2$ is $\frac{1}{4}$.

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. What is the rule for dividing powers with the same base?
2. How do you express numbers like 16 and 32 as powers of 2?
3. How can negative exponents be interpreted?
4. How do you handle division of exponents with different bases?
5. What is the result when raising a power to another power?

Tip: When simplifying expressions with powers, always look for common bases to apply exponent rules more easily.