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

A. [tex]\sqrt[7]{4 x}[/tex]
B. [tex]4 \sqrt[2]{x}[/tex]
C. [tex]x^{\frac{4}{7}}[/tex]
D. [tex]x^{\frac{7}{4}}[/tex]



Answer :

To simplify the expression [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex], we will use the properties of exponents.

1. First, recognize that [tex]\(\sqrt[7]{x}\)[/tex] can be written as [tex]\(x^{\frac{1}{7}}\)[/tex].
2. Rewrite the given expression using this exponent form:
[tex]\[ \sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x} = x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \][/tex]
3. Next, use the property of exponents that states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]. Add the exponents together:
[tex]\[ x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} \cdot x^{\frac{1}{7}} = x^{\frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7}} \][/tex]
4. Simplify the sum of the exponents:
[tex]\[ \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7} = \frac{4}{7} \][/tex]
5. Therefore, the expression simplifies to:
[tex]\[ x^{\frac{4}{7}} \][/tex]

So, the simplified form of [tex]\(\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}\)[/tex] is [tex]\(x^{\frac{4}{7}}\)[/tex].

The correct answer is:
[tex]\[ x^{\frac{4}{7}} \][/tex]