Question
What is the simplified base for the function \( f(x)=2\left(\sqrt{27}^{2 x}\right) \) ? 3 9 18 9
Ask by Frank May. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified base for the function \( f(x) = 2\left(\sqrt{27}^{2x}\right) \) is **3**.
Solution
To simplify the base of the function \( f(x) = 2\left(\sqrt{27}^{2x}\right) \), follow these steps:
1. **Simplify \(\sqrt{27}\):**
\[
\sqrt{27} = 27^{1/2} = (3^3)^{1/2} = 3^{3/2}
\]
2. **Raise \(\sqrt{27}\) to the power of \(2x\):**
\[
\left(\sqrt{27}\right)^{2x} = \left(3^{3/2}\right)^{2x} = 3^{3x}
\]
3. **Express the function with the simplified base:**
\[
f(x) = 2 \cdot 3^{3x}
\]
Here, the base is \(3\).
So, the **simplified base** for the function \( f(x) = 2\left(\sqrt{27}^{2x}\right) \) is **3**.
**Answer:** 3
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Simplify this solution The Deep Dive
To simplify \( f(x) = 2(\sqrt{27}^{2x}) \), we start by rewriting \( \sqrt{27} \) as \( 27^{1/2} \). This gives us \( f(x) = 2((27^{1/2})^{2x}) \). Using the power of a power property, we simplify this to \( f(x) = 2(27^{x}) \). Because \( 27 = 3^3 \), we can express it as \( 27^x = (3^3)^x = 3^{3x} \). Therefore, the function simplifies to \( f(x) = 2 \cdot 3^{3x} \). The base here is 3, which matches the requirement for simplification. Hence, the answer is \( 3 \).
