5. MP Use Tools Write an expression with a single exponent that is equivalent to \( 8^{-4} \cdot\left(8^{2}\right)^{4} \). State what strategy and tool you will use to answer the question, explain your choice, and then find the answer.

To solve the expression \( 8^{-4} \cdot (8^{2})^{4} \) and write it with a single exponent, we can use the properties of exponents. ### Known Conditions: 1. The expression involves the base \( 8 \) and different exponents. 2. We can apply the exponent multiplication rule, which states that \( (a^m)^n = a^{m \cdot n} \), and the multiplication rule, which states that \( a^m \cdot a^n = a^{m+n} \). ### Strategy: 1. First, simplify \( (8^{2})^{4} \) using the exponent multiplication rule. 2. Then, combine the results using the multiplication rule for exponents. ### Steps: 1. Simplify \( (8^{2})^{4} \): \[ (8^{2})^{4} = 8^{2 \cdot 4} = 8^{8} \] 2. Now substitute back into the original expression: \[ 8^{-4} \cdot 8^{8} \] 3. Apply the multiplication rule: \[ 8^{-4 + 8} = 8^{4} \] Now, we have simplified the expression to a single exponent. ### Final Answer: The expression \( 8^{-4} \cdot (8^{2})^{4} \) is equivalent to \( 8^{4} \). I will now use the tool to confirm the calculations. Calculate the value by following steps: - step0: Calculate: \(8^{-4}\left(8^{2}\right)^{4}\) - step1: Multiply the exponents: \(8^{-4}\times 8^{2\times 4}\) - step2: Multiply the numbers: \(8^{-4}\times 8^{8}\) - step3: Multiply: \(8^{-4+8}\) - step4: Add the numbers: \(8^{4}\) - step5: Evaluate the power: \(4096\) The calculation confirms that the expression \( 8^{-4} \cdot (8^{2})^{4} \) simplifies to \( 8^{4} \), which is equal to \( 4096 \). Thus, the final answer is: \[ 8^{-4} \cdot (8^{2})^{4} = 8^{4} = 4096 \]