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Select the correct answer.

Rewrite the following radical expression in rational exponent form:

[tex]\[ \left(\sqrt[7]{x}\right)^3 \][/tex]

A. [tex]\(\left(\frac{1}{x^3}\right)^7\)[/tex]

B. [tex]\(x^{\frac{7}{3}}\)[/tex]

C. [tex]\(x^{\frac{3}{7}}\)[/tex]

D. [tex]\(\frac{x^3}{x^7}\)[/tex]


Sagot :

To rewrite the given radical expression [tex]\((\sqrt[7]{x})^3\)[/tex] in rational exponent form, follow these steps:

1. Understand the radical notation and the n-th root:
- The n-th root of a number [tex]\(x\)[/tex] can be expressed as [tex]\(x^{1/n}\)[/tex].
- Here, the 7-th root of [tex]\(x\)[/tex] is [tex]\(\sqrt[7]{x}\)[/tex], which can be written as [tex]\(x^{1/7}\)[/tex].

2. Apply the power to the radical expression:
- The given expression is [tex]\((\sqrt[7]{x})^3\)[/tex].
- As mentioned, [tex]\(\sqrt[7]{x} = x^{1/7}\)[/tex].

3. Use the properties of exponents:
- When raising a power to another power, you multiply the exponents.
- Therefore, [tex]\((x^{1/7})^3\)[/tex] equals [tex]\(x^{(1/7) \cdot 3}\)[/tex].

4. Simplify the exponent:
- Multiply the exponents: [tex]\(\frac{1}{7} \times 3 = \frac{3}{7}\)[/tex].

5. Combine the steps:
- The expression [tex]\((\sqrt[7]{x})^3\)[/tex] simplifies to [tex]\(x^{3/7}\)[/tex].

Thus, the correct answer is:
[tex]\[ x^{\frac{3}{7}} \][/tex]