Let's simplify the given expression step by step:

The expression is:

$\frac{2^5 \times 16 \times 2^{-4}}{4^2 \times 2^5 \times 32}$

Step 1: Rewrite the constants as powers of 2

• $16 = 2^4$
• $4^2 = (2^2)^2 = 2^4$
• $32 = 2^5$

Substituting these values into the expression, we get:

$\frac{2^5 \times 2^4 \times 2^{-4}}{2^4 \times 2^5 \times 2^5}$

Step 2: Simplify the powers of 2

Using the property of exponents $a^m \times a^n = a^{m+n}$ and $\frac{a^m}{a^n} = a^{m-n}$:

In the numerator:

$2^5 \times 2^4 \times 2^{-4} = 2^{5 + 4 - 4} = 2^5$

In the denominator:

$2^4 \times 2^5 \times 2^5 = 2^{4 + 5 + 5} = 2^{14}$

Now the expression becomes:

$\frac{2^5}{2^{14}}$

Step 3: Final simplification

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

$2^{5-14} = 2^{-9}$

Therefore, the simplified expression is:

$\boxed{2^{-9}}$

This is the simplified result, or equivalently:

$\boxed{\frac{1}{512}}$

Would you like further clarification or details?

Here are 5 related questions to deepen your understanding:

1. How do exponent rules apply when multiplying and dividing powers of the same base?
2. What is the importance of negative exponents in simplifying expressions?
3. How do we simplify expressions involving both negative and positive exponents?
4. What would change in the problem if the base was not 2 but another number?
5. How do exponent properties apply when dealing with fractional bases?

Tip: When simplifying powers, always look for ways to express numbers as powers of a common base before applying exponent rules.