Which expression is equivalent to [tex]$\sqrt[3]{27x} + \sqrt[3]{x}[tex]$[/tex], if [tex]$[/tex]x \neq 0$[/tex]?

A. [tex]$4 \sqrt[3]{x}$[/tex]
B. [tex]$\sqrt[3]{28x}$[/tex]
C. [tex]$3 \sqrt[3]{x}$[/tex]
D. [tex]$4 \sqrt[3]{x^2}$[/tex]



Answer :

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]