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Which of the following is the simplified form of [tex]$\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}$[/tex]?

A. [tex]x^{\frac{3}{7}}[/tex]

B. [tex]x^{\frac{1}{7}}[/tex]

C. [tex]x^{\frac{3}{21}}[/tex]

D. [tex]21 \sqrt{x}[/tex]


Sagot :

Sure, let's break this down step-by-step:

First, we start with the expression [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex].

1. Express each term using exponents:
[tex]\[ \sqrt[7]{x} = x^{\frac{1}{7}} \][/tex]
Thus, our expression [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] can be written as:
[tex]\[ x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \][/tex]

2. Combine the exponents:
When we multiply terms with the same base, we add the exponents. We have:
[tex]\[ x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} = x^{\left(\frac{1}{7} + \frac{1}{7} + \frac{1}{7}\right)} \][/tex]

3. Add the exponents:
[tex]\[ \frac{1}{7} + \frac{1}{7} + \frac{1}{7} = \frac{3}{7} \][/tex]

4. Write the final expression:
[tex]\[ x^{\frac{3}{7}} \][/tex]

So, the simplified form of [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] is [tex]\(\boxed{x^{\frac{3}{7}}}\)[/tex].