Question

1. simplify the expression.

$(4e^{-2x})^3 = \square$

Explanation:

Step1: Apply power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). For the expression \((4e^{-2x})^3\), we can apply this rule as follows:
\((4e^{-2x})^3 = 4^3 \cdot (e^{-2x})^3\)
Calculating \(4^3\), we get \(4^3 = 64\).

Step2: Apply power of a power rule

The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). For the term \((e^{-2x})^3\), we apply this rule:
\((e^{-2x})^3 = e^{-2x \cdot 3} = e^{-6x}\)

Step3: Combine the results

Now, we combine the results from Step 1 and Step 2:
\(4^3 \cdot (e^{-2x})^3 = 64 \cdot e^{-6x} = 64e^{-6x}\)

Answer:

\(64e^{-6x}\)