Let's simplify the given function step-by-step to identify the base.
Given function:
[tex]\[ f(x) = 2\left(\sqrt[3]{27^{2x}}\right) \][/tex]
First, we need to simplify the expression inside the parentheses:
[tex]\[ \sqrt[3]{27^{2x}} \][/tex]
Recall that the cube root of a number can be expressed as a power of [tex]\( \frac{1}{3} \)[/tex]:
[tex]\[ \sqrt[3]{27^{2x}} = \left(27^{2x}\right)^{\frac{1}{3}} \][/tex]
We can use the power rule of exponents to simplify this expression:
[tex]\[ \left(27^{2x}\right)^{\frac{1}{3}} = 27^{\frac{2x}{3}} \][/tex]
Next, recognize that 27 can be rewritten as a power of 3:
[tex]\[ 27 = 3^3 \][/tex]
Now substitute [tex]\( 3^3 \)[/tex] for 27:
[tex]\[ 27^{\frac{2x}{3}} = (3^3)^{\frac{2x}{3}} \][/tex]
Using the power rule for exponents again, which states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ (3^3)^{\frac{2x}{3}} = 3^{3 \cdot \frac{2x}{3}} = 3^{2x} \][/tex]
Thus, the expression inside the parentheses simplifies to [tex]\( 3^{2x} \)[/tex].
Now, let's put this back into the original function:
[tex]\[ f(x) = 2 \cdot 3^{2x} \][/tex]
Notice that in the function [tex]\( f(x) = 2 \cdot 3^{2x} \)[/tex], the base of the exponential part is [tex]\( 3 \)[/tex].
Therefore, the simplified base of the function is:
[tex]\[ \boxed{3} \][/tex]
