Select the equivalent expression.

Question

\[\left(6^{-4}\cdot 8^{-7}\right)^{-9}=?\]
Choose 1 answer:
Choose 1 answer:
(Choice A)

\[\dfrac{1}{6^{13}\cdot 8^{16}}\]
A

\[\dfrac{1}{6^{13}\cdot 8^{16}}\]
(Choice B)
\[6^{5}\cdot 8^{2}\]
B
\[6^{5}\cdot 8^{2}\]
(Choice C)
\[6^{36}\cdot 8^{63}\]
C
\[6^{36}\cdot 8^{63}\]

Answer 1

To solve the expression \(\left(6^{-4}\cdot 8^{-7}\right)^{-9}\), we can use the properties of exponents.

1. Apply the exponent rule \((a \cdot b)^m = a^m \cdot b^m\): \[ \left(6^{-4} \cdot 8^{-7}\right)^{-9} = \left(6^{-4}\right)^{-9} \cdot \left(8^{-7}\right)^{-9} \]

2. Apply the exponent rule \(a^{-m} = \frac{1}{a^m}\) and also \(a^{m \cdot n} = a^{m \cdot n}\): \[ \left(6^{-4}\right)^{-9} = 6^{36} \quad \text{and} \quad \left(8^{-7}\right)^{-9} = 8^{63} \]

3. Therefore, we can combine these: \[ 6^{36} \cdot 8^{63} \]

So, the final equivalent expression is: \[ 6^{36} \cdot 8^{63} \]

Thus, the answer is Choice C: \[6^{36} \cdot 8^{63}\].