Questions: Select the equivalent expression. (6^(-4) * 8^(-7))^(-9)=? Choose 1 answer: (A) 6^5 * 8^2 (B) 1/(6^13 * 8^16) (C) 6^36 * 8^63

Solution Steps

To solve the given expression \(\left(6^{-4} \cdot 8^{-7}\right)^{-9}\), we need to apply the properties of exponents. Specifically, we use the power of a product property \((a^m \cdot b^n)^p = a^{m \cdot p} \cdot b^{n \cdot p}\) and the power of a power property \((a^m)^n = a^{m \cdot n}\). First, distribute the exponent \(-9\) to both terms inside the parentheses, and then simplify the resulting exponents.

To simplify the expression \(\left(6^{-4} \cdot 8^{-7}\right)^{-9}\), we first apply the power of a product property: \[ (a^m \cdot b^n)^p = a^{m \cdot p} \cdot b^{n \cdot p} \] This gives us: \[ 6^{-4 \cdot (-9)} \cdot 8^{-7 \cdot (-9)} \]

Next, we calculate the new exponents: \[ -4 \cdot (-9) = 36 \] \[ -7 \cdot (-9) = 63 \] Thus, the expression simplifies to: \[ 6^{36} \cdot 8^{63} \]

The expression \(6^{36} \cdot 8^{63}\) is a very large number. However, we are only interested in matching it to one of the given choices. The expression matches option (C) \(6^{36} \cdot 8^{63}\).

Final Answer