To solve the equation and simplify the expression [tex]\(2 \sqrt[3]{2 x} \cdot \sqrt[3]{x}\)[/tex], we will use the properties of exponents and radicals.
First, rewrite the given expression [tex]\(2 \sqrt[3]{2 x} \cdot \sqrt[3]{x}\)[/tex] using exponent notation. Recall that the cube root of [tex]\(a\)[/tex] is [tex]\(a^\frac{1}{3}\)[/tex]:
[tex]\[
2 \sqrt[3]{2x} \cdot \sqrt[3]{x} = 2 (2x)^\frac{1}{3} \cdot x^\frac{1}{3}
\][/tex]
Next, apply the property of exponents that states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]. In this case, both terms inside the cube roots have the same base [tex]\(x\)[/tex]:
[tex]\[
(2x)^\frac{1}{3} \cdot x^\frac{1}{3} = 2^\frac{1}{3} x^\frac{1}{3} \cdot x^\frac{1}{3}
\][/tex]
Combine the exponents for [tex]\(x\)[/tex]:
[tex]\[
(2x)^\frac{1}{3} \cdot x^\frac{1}{3} = 2^\frac{1}{3} x^{\frac{1}{3} + \frac{1}{3}} = 2^\frac{1}{3} x^\frac{2}{3}
\][/tex]
Now, multiply this by the constant 2:
[tex]\[
2 \cdot 2^\frac{1}{3} x^\frac{2}{3}
\][/tex]
Recall that [tex]\(2 = 2^1\)[/tex], so we can combine the factors of 2:
[tex]\[
2^1 \cdot 2^\frac{1}{3} = 2^{1+\frac{1}{3}} = 2^\frac{4}{3}
\][/tex]
This gives:
[tex]\[
2^\frac{4}{3} x^\frac{2}{3}
\][/tex]
Finally, express this in radical form. The exponent [tex]\(\frac{4}{3}\)[/tex] represents the cube root of the base raised to the power 4:
[tex]\[
(2^4 x^2)^\frac{1}{3} = (16 x^2)^\frac{1}{3}
\][/tex]
Therefore, the expression [tex]\(2 \sqrt[3]{2 x} \cdot \sqrt[3]{x}\)[/tex] is equivalent to [tex]\(\sqrt[3]{16 x^2}\)[/tex]. This corresponds to choice K:
[tex]\[
\boxed{\sqrt[3]{16 x^2}}
\][/tex]
