Let's go through each of the incorrect steps (parts d and e) and correct them:

(d) $(2x^{-3})^4x^{-4}$

We need to simplify the expression using the laws of exponents.

1. Apply the power rule to $(2x^{-3})^4$: $(2x^{-3})^4 = 2^4 \cdot (x^{-3})^4 = 16 \cdot x^{-12}$

2. Now combine this result with $x^{-4}$: $16 \cdot x^{-12} \cdot x^{-4} = 16 \cdot x^{-12 - 4} = 16 \cdot x^{-16}$

So, the simplified expression is: $\boxed{16x^{-16}}$ You had $16x^8$, which is incorrect because the exponents were added incorrectly.

(e) $\frac{4a^{7/2} \cdot a^{1/2}}{a^{-1}}$

1. First, simplify the numerator by combining the powers of $a$: $a^{7/2} \cdot a^{1/2} = a^{(7/2 + 1/2)} = a^{8/2} = a^4$

2. Now simplify the entire expression by dividing by $a^{-1}$: $\frac{4 \cdot a^4}{a^{-1}} = 4 \cdot a^{4 - (-1)} = 4 \cdot a^{4 + 1} = 4 \cdot a^5$

So, the correct simplified expression is: $\boxed{4a^5}$

You had $\sqrt{16384a^8} \cdot a$, which seems to have misunderstood the exponents.

Let me know if you need more details on any of these steps! Here are 5 related questions to help expand your understanding:

1. How do you simplify expressions with negative exponents?
2. What is the power of a product rule, and how can it be applied to simplify expressions like $(2x^{-3})^4$?
3. How do you simplify the square root of expressions with exponents?
4. What are the key rules to remember when dividing powers with the same base?
5. How would you solve an expression like $(a^2b^{-3})^3$?

Tip:

Always carefully apply the laws of exponents, particularly when dealing with negative exponents and products.