To simplify the expression $\dfrac{4^{-3}}{4^{-1}}$, we can use the property of exponents:

$\dfrac{a^m}{a^n} = a^{m-n}$

So applying this property to your expression:

$\dfrac{4^{-3}}{4^{-1}} = 4^{-3 - (-1)} = 4^{-3 + 1} = 4^{-2}$

Thus, the simplified result is:

$4^{-2}$

You can rewrite this as:

$4^{-2} = \dfrac{1}{4^2} = \dfrac{1}{16}$

So, the final answer is:

$\dfrac{1}{16}$

Would you like to dive into how this works in more detail? Or do you have any other questions?

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Here are some related questions for you:

1. How do you simplify other expressions with negative exponents, like $\dfrac{2^{-4}}{2^{-2}}$?
2. What is the result when you multiply two terms with the same base and negative exponents?
3. Can you explain why $4^{-2}$ is the same as $\dfrac{1}{16}$?
4. How would you handle the expression $\dfrac{5^{-1}}{5^{-3}}$?
5. What are some real-world applications of using negative exponents?

Tip: Negative exponents represent the reciprocal of the base raised to the positive exponent, which is useful for simplifying fractions.