We are asked to simplify the given mathematical expression:

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

Step 1: Rewrite the constants as powers of 2

• $16 = 2^4$
• $4 = 2^2$
• $32 = 2^5$

So the expression becomes: $\frac{2^5 \times 2^4 \times 2^{-4}}{2^2 \times 2^5 \times 2^5}$

Step 2: Combine powers of 2 using the rule $a^m \times a^n = a^{m+n}$

In the numerator: $2^5 \times 2^4 \times 2^{-4} = 2^{5 + 4 - 4} = 2^5$

In the denominator: $2^2 \times 2^5 \times 2^5 = 2^{2 + 5 + 5} = 2^{12}$

Now the expression becomes: $\frac{2^5}{2^{12}}$

Step 3: Simplify the powers of 2 using the rule $\frac{a^m}{a^n} = a^{m-n}$

$\frac{2^5}{2^{12}} = 2^{5 - 12} = 2^{-7}$

Step 4: Convert the negative exponent to a fraction

$2^{-7} = \frac{1}{2^7} = \frac{1}{128}$

Final Answer:

The simplified form of the given expression is: $\frac{1}{128}$

Would you like further explanation on any step?

Here are 5 related questions for practice:

1. Simplify $\frac{3^4 \times 9}{27}$.
2. Simplify $\frac{5^3 \times 25}{5^5}$.
3. Simplify $\frac{2^6 \times 8}{16 \times 4}$.
4. Simplify $\frac{3^2 \times 81}{9^2}$.
5. Simplify $\frac{2^4 \times 32}{16 \times 2^2}$.

Tip: When simplifying exponents, always try to express constants as powers of the same base to make calculations easier.