The expression provided in the image is:

$\frac{4^{2n} \times 8^{3n}}{16^{4n}}$

To simplify this expression, let’s work through each part step-by-step using properties of exponents.

Step 1: Rewrite terms with a common base

1. Rewrite $4^{2n}$: Since $4 = 2^2$, we can express $4^{2n}$ as: $4^{2n} = (2^2)^{2n} = 2^{4n}$

2. Rewrite $8^{3n}$: Since $8 = 2^3$, we can express $8^{3n}$ as: $8^{3n} = (2^3)^{3n} = 2^{9n}$

3. Rewrite $16^{4n}$: Since $16 = 2^4$, we can express $16^{4n}$ as: $16^{4n} = (2^4)^{4n} = 2^{16n}$

Step 2: Substitute and Simplify the Expression

Now, substitute these expressions back into the original problem:

$\frac{4^{2n} \times 8^{3n}}{16^{4n}} = \frac{2^{4n} \times 2^{9n}}{2^{16n}}$

Step 3: Apply the Product and Quotient Rules of Exponents

Combine the terms in the numerator using the product rule of exponents $(a^m \times a^n = a^{m+n})$:

$\frac{2^{4n} \times 2^{9n}}{2^{16n}} = \frac{2^{4n + 9n}}{2^{16n}} = \frac{2^{13n}}{2^{16n}}$

Now, use the quotient rule of exponents $\frac{a^m}{a^n} = a^{m-n}$:

$2^{13n - 16n} = 2^{-3n}$

Final Answer

The simplified form of the expression is:

$2^{-3n} = \frac{1}{2^{3n}}$

Would you like more details on any step or have further questions?

Here are some related questions to expand your understanding:

1. How would you simplify $\frac{3^{2n} \times 9^{n}}{27^{3n}}$?
2. Can you apply similar steps to simplify $\frac{5^{3n} \times 25^{2n}}{125^{4n}}$?
3. How do exponent rules apply when bases are different but the powers are the same?
4. How would the solution change if we had $16^{2n}$ in the denominator instead of $16^{4n}$?
5. What happens to the answer if $n$ is negative in this problem?

Tip: Always try to express terms with a common base to simplify expressions involving exponents.