Select the equivalent expression.
 
\[\left(\dfrac{b^{7}}{4^{5}}\right)^{-3}=?\]
Choose 1 answer:
Choose 1 answer:
(Choice A)   
\[{b^{-21}}\cdot{4^{-15}}\]
A
\[{b^{-21}}\cdot{4^{-15}}\]
(Choice B)   
 
\[\dfrac{b^{21}}{4^{15}}\]
B
 
\[\dfrac{b^{21}}{4^{15}}\]
(Choice C)   
 
\[\dfrac{4^{15}}{b^{21}}\]
C
 
\[\dfrac{4^{15}}{b^{21}}\]

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!

Select the equivalent expression. 
\[\left(\dfrac{b^{7}}{4^{5}}\right)^{-3}=?\]

Learn how to simplify the expression \(\left( \dfrac{b^{7}}{4^{5}} \right)^{-3}\) using the negative exponent rule. This problem covers the application of the negative exponent rule to fractions. Suitable for students in Grades 7-9.

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