To determine which expression is equivalent to [tex]\(\sqrt[3]{27x} + \sqrt[3]{x}\)[/tex], let's break it down step-by-step.
1. Expression Analysis:
We start with the expression:
[tex]\[
\sqrt[3]{27x} + \sqrt[3]{x}
\][/tex]
2. Simplify [tex]\(\sqrt[3]{27x}\)[/tex]:
Notice that [tex]\(27\)[/tex] is a perfect cube, specifically [tex]\(27 = 3^3\)[/tex]. Therefore:
[tex]\[
\sqrt[3]{27x} = \sqrt[3]{3^3 \cdot x} = 3 \cdot \sqrt[3]{x}
\][/tex]
This follows from the property of cube roots: [tex]\(\sqrt[3]{a^3 \cdot b} = a \cdot \sqrt[3]{b}\)[/tex].
3. Combine the Simplified Terms:
Now substitute back into the original expression:
[tex]\[
\sqrt[3]{27x} + \sqrt[3]{x} = 3 \cdot \sqrt[3]{x} + \sqrt[3]{x}
\][/tex]
4. Factor the Expression:
Factor out [tex]\(\sqrt[3]{x}\)[/tex] from both terms:
[tex]\[
3 \cdot \sqrt[3]{x} + \sqrt[3]{x} = \left(3 + 1\right) \cdot \sqrt[3]{x} = 4 \cdot \sqrt[3]{x}
\][/tex]
Thus, the expression equivalent to [tex]\(\sqrt[3]{27x} + \sqrt[3]{x}\)[/tex] is:
[tex]\[
4 \cdot \sqrt[3]{x}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{A \; 4 \sqrt[3]{x}}
\][/tex]
