To determine which expression is equivalent to the given expression \(\sqrt[3]{27 x} + \sqrt[3]{x}\) where \(x \neq 0\), let's start by analyzing and simplifying the original expression step-by-step.
1. Identify the cube roots individually:
The expression is \(\sqrt[3]{27 x} + \sqrt[3]{x}\).
2. Simplify \(\sqrt[3]{27 x}\):
Notice that \(27\) can be expressed as \(3^3\), so:
[tex]\[
\sqrt[3]{27 x} = \sqrt[3]{3^3 x} = 3 \sqrt[3]{x}
\][/tex]
3. Substitute the simplified form back into the expression:
Now, replace \(\sqrt[3]{27 x}\) with \(3 \sqrt[3]{x}\) in the original expression. This gives us:
[tex]\[
3 \sqrt[3]{x} + \sqrt[3]{x}
\][/tex]
4. Combine like terms in the expression:
Both terms \(\sqrt[3]{x}\) are like terms, so you can add their coefficients together:
[tex]\[
3 \sqrt[3]{x} + \sqrt[3]{x} = (3 + 1) \sqrt[3]{x} = 4 \sqrt[3]{x}
\][/tex]
Therefore, the expression \(\sqrt[3]{27 x} + \sqrt[3]{x}\) simplifies to \(4 \sqrt[3]{x}\).
The correct answer is:
A. [tex]\(4 \sqrt[3]{x}\)[/tex]
