We are tasked with simplifying the expression:

$\left(\dfrac{b^{7}}{4^{5}}\right)^{-3}$

Step 1: Apply the negative exponent rule

The negative exponent rule states that:

$\left(\dfrac{a}{b}\right)^{-n} = \dfrac{b^{n}}{a^{n}}$

So we can rewrite the expression as:

$\left(\dfrac{b^{7}}{4^{5}}\right)^{-3} = \dfrac{4^{5 \cdot 3}}{b^{7 \cdot 3}}$

Step 2: Simplify the exponents

Now, calculate the new exponents:

$5 \cdot 3 = 15 \quad \text{and} \quad 7 \cdot 3 = 21$

So the expression becomes:

$\dfrac{4^{15}}{b^{21}}$

Step 3: Match with the choices

We see that the expression simplifies to:

$\dfrac{4^{15}}{b^{21}}$

This matches Choice C.

Final Answer:

$\boxed{\text{Choice C: } \dfrac{4^{15}}{b^{21}}}$

Would you like more details on how this works, or have any further questions?

Here are 5 related questions for practice:

1. Simplify $\left( \dfrac{a^{3}}{b^{4}} \right)^{-2}$.
2. Simplify $\left( \dfrac{x^{5}}{y^{2}} \right)^{-3}$.
3. What happens when you raise a fraction to a negative exponent?
4. Simplify $\left( \dfrac{2x^{3}}{5y^{2}} \right)^{-4}$.
5. How do you simplify expressions involving both negative and fractional exponents?

Tip: When dealing with negative exponents, always flip the fraction and change the sign of the exponent!