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]
