Question

Select the equivalent expression.

\[\left(\dfrac{3^{-6}}{7^{-3}}\right)^{5}=?\]
Choose 1 answer:
Choose 1 answer:
(Choice A)

\[\dfrac{7^{15}}{3^{30}}\]
A

\[\dfrac{7^{15}}{3^{30}}\]
(Choice B)

\[\dfrac{7^{3}}{3^{-30}}\]
B

\[\dfrac{7^{3}}{3^{-30}}\]
(Choice C)

\[\dfrac{3^{15}}{7^{30}}\]
C

\[\dfrac{3^{15}}{7^{30}}\]

Answer 1

To simplify the expression \(\left(\dfrac{3^{-6}}{7^{-3}}\right)^{5}\), we will follow these steps:

1. Simplify the inner fraction:
\[
\dfrac{3^{-6}}{7^{-3}} = 3^{-6} \cdot 7^{3} = \dfrac{7^{3}}{3^{6}}.
\]

2. Now raise the fraction to the power of 5:
\[
\left(\dfrac{7^{3}}{3^{6}}\right)^{5} = \dfrac{(7^{3})^{5}}{(3^{6})^{5}}.
\]

3. Apply the power rule \((a^{m})^{n} = a^{m \cdot n}\):
\[
(7^{3})^{5} = 7^{15} \quad \text{and} \quad (3^{6})^{5} = 3^{30}.
\]

4. Therefore, the expression becomes:
\[
\dfrac{7^{15}}{3^{30}}.
\]

Now, let's compare this expression with the given choices:

- (Choice A): \(\dfrac{7^{15}}{3^{30}}\)
- (Choice B): \(\dfrac{7^{3}}{3^{-30}}\)
- (Choice C): \(\dfrac{3^{15}}{7^{30}}\)

Since we have determined that the simplified expression is \(\dfrac{7^{15}}{3^{30}}\), the equivalent expression is:

**Choice A: \(\dfrac{7^{15}}{3^{30}}\)**.