Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sqrt[3]{8 e^{12x}} = 2e^{4x}
\]
Solution
To simplify the expression \( \sqrt[3]{8 e^{12 x}} \), follow these steps:
1. **Break Down the Cube Root:**
The cube root of a product can be separated into the product of the cube roots:
\[
\sqrt[3]{8 e^{12x}} = \sqrt[3]{8} \cdot \sqrt[3]{e^{12x}}
\]
2. **Simplify Each Component:**
- **Cube Root of 8:**
\[
\sqrt[3]{8} = 2 \quad \text{since} \quad 2^3 = 8
\]
- **Cube Root of \( e^{12x} \):**
\[
\sqrt[3]{e^{12x}} = e^{\frac{12x}{3}} = e^{4x}
\]
3. **Combine the Simplified Components:**
\[
\sqrt[3]{8 e^{12x}} = 2 \cdot e^{4x} = 2e^{4x}
\]
**Final Simplified Form:**
\[
\sqrt[3]{8 e^{12x}} = 2e^{4x}
\]
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Simplify this solution Bonus Knowledge
To simplify \( \sqrt[3]{8 e^{12 x}} \), we can break it down into its components. The cube root of 8 is \( 2 \), since \( 2^3 = 8 \). Next, for \( e^{12x} \), we apply the power property of exponents: \( \sqrt[3]{e^{12x}} = e^{(12x)/3} = e^{4x} \). Putting it all together, we have: \[ \sqrt[3]{8 e^{12 x}} = 2 e^{4 x} \] And there you have it, neatly simplified!
