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Select the correct answer.

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


Sagot :

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]
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