Question

Select the equivalent expression.
\[\left(3^{-8}\cdot 7^{3}\right)^{-2}=?\]
Choose 1 answer:
Choose 1 answer:
(Choice A)
\[21^{10}\]
A
\[21^{10}\]
(Choice B)
\[3^{16}\cdot 7^{-6}\]
B
\[3^{16}\cdot 7^{-6}\]
(Choice C)

\[\dfrac{7^6}{3^{16}}\]
C

\[\dfrac{7^6}{3^{16}}\]

Answer 1

To simplify the expression \(\left(3^{-8}\cdot 7^{3}\right)^{-2}\), we start by applying the property of exponents that states \((a \cdot b)^{-n} = a^{-n} \cdot b^{-n}\).

1. Apply the exponent to each factor inside the parentheses:
\[
\left(3^{-8}\cdot 7^{3}\right)^{-2} = \left(3^{-8}\right)^{-2} \cdot \left(7^{3}\right)^{-2}
\]

2. Simplify each term by applying the power of a power property, which states \((a^m)^n = a^{m \cdot n}\):
\[
\left(3^{-8}\right)^{-2} = 3^{-8 \cdot -2} = 3^{16}
\]
\[
\left(7^{3}\right)^{-2} = 7^{3 \cdot -2} = 7^{-6}
\]

3. Now combine the results from the above steps:
\[
3^{16} \cdot 7^{-6}
\]

Now we check which option matches our result:

- (Choice A) \(21^{10}\)
- (Choice B) \(3^{16}\cdot 7^{-6}\)
- (Choice C) \(\dfrac{7^6}{3^{16}}\)

The equivalent expression is:
\[
\boxed{B} \quad 3^{16} \cdot 7^{-6}
\]