To simplify the function [tex]\( f(x) = 2\left(\sqrt[3]{27^{2x}}\right) \)[/tex], let's go through a step-by-step process:
1. Simplify the Base:
- Recall that 27 can be written as [tex]\( 3^3 \)[/tex]. So we rewrite [tex]\( 27^{2x} \)[/tex]:
[tex]\[
27^{2x} = (3^3)^{2x}
\][/tex]
2. Apply Exponent Rules:
- Using the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we simplify [tex]\((3^3)^{2x}\)[/tex]:
[tex]\[
(3^3)^{2x} = 3^{3 \cdot 2x} = 3^{6x}
\][/tex]
3. Simplify the Inner Function:
- Now we need to find [tex]\(\sqrt[3]{3^{6x}}\)[/tex]:
[tex]\[
\sqrt[3]{3^{6x}} = (3^{6x})^{1/3}
\][/tex]
- Again, applying the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[
(3^{6x})^{1/3} = 3^{6x \cdot 1/3} = 3^{2x}
\][/tex]
4. Combine the Results:
- Substituting back, we get:
[tex]\[
f(x) = 2 \left( 3^{2x} \right)
\][/tex]
Therefore, the simplified base of the function [tex]\( f(x) = 2\left(\sqrt[3]{27^{2x}}\right) \)[/tex] is [tex]\(3\)[/tex].
[tex]\[
\boxed{3}
\][/tex]
