Let's solve the problem from the image. The problem is asking for the simplest form of the given expression:

$\frac{2^{3(n+1)} \times (2^n - 1)}{(2^{n-1})}$

Simplifying the Expression:

1. Simplify the numerator: The numerator is $2^{3(n+1)} \times (2^n - 1)$.

   • Expand $2^{3(n+1)}$: $2^{3(n+1)} = 2^{3n + 3} = 2^{3n} \times 2^3 = 2^{3n} \times 8$
   • So the numerator becomes: $8 \times 2^{3n} \times (2^n - 1)$

2. Simplify the denominator: The denominator is $2^{n-1}$.

   • This can be rewritten as: $2^{n-1} = \frac{2^n}{2}$

3. Combine the expression: $\frac{8 \times 2^{3n} \times (2^n - 1)}{2^{n-1}} = \frac{8 \times 2^{3n} \times (2^n - 1)}{\frac{2^n}{2}}$

   • Simplify further: $= \frac{8 \times 2^{3n} \times (2^n - 1) \times 2}{2^n} = \frac{16 \times 2^{3n} \times (2^n - 1)}{2^n}$
   • Simplify the powers of 2: $= 16 \times 2^{2n} \times (2^n - 1)$
   • The expression simplifies to: $16 \times (2^n \times 2^n \times (2^n - 1)) = 16 \times 2^{2n} \times (2^n - 1)$

Thus, the final simplified form would be:

$\boxed{16 \times 2^{2n} \times (2^n - 1)}$

Let's check the multiple-choice options for the answer.

Answer Options:

• a. $2$
• b. $2^2$
• c. $2^{2n}$
• d. $2^{2n-1}$

The correct answer appears to align with option c. $2^{2n}$, but considering the possible presence of multiple forms, the simplest and closest match is 16 \times 2^{2n}.

Do you want details or have any questions about this solution? Here are 5 related questions to further your understanding:

1. What is the general strategy for simplifying expressions involving exponents?
2. How does multiplying and dividing powers of the same base work?
3. Can you explain the difference between factorizing and expanding expressions?
4. Why is it useful to simplify mathematical expressions?
5. How would the expression change if the base number was not 2?

Tip: Always check if the terms in the numerator and denominator share a common factor that can simplify the expression.