To rewrite the radical expression [tex]\(\sqrt[7]{x^3}\)[/tex] as an expression with a rational exponent, follow these steps:
1. Identify the form of the radical expression: The given expression is [tex]\(\sqrt[7]{x^3}\)[/tex], where [tex]\(7\)[/tex] is the index of the root and [tex]\(3\)[/tex] is the exponent of [tex]\(x\)[/tex].
2. Understand the relationship between radicals and exponents: A radical expression of the form [tex]\(\sqrt[n]{x^m}\)[/tex] can be rewritten using a rational exponent as [tex]\(x^{m/n}\)[/tex].
3. Substitute the values into the relationship:
- [tex]\(m\)[/tex] is the exponent of the radicand, which is [tex]\(3\)[/tex] in this case.
- [tex]\(n\)[/tex] is the index of the root, which is [tex]\(7\)[/tex] in this case.
4. Apply the relationship: Substitute [tex]\(m = 3\)[/tex] and [tex]\(n = 7\)[/tex] into the expression [tex]\(x^{m/n}\)[/tex] to get [tex]\(x^{3/7}\)[/tex].
Therefore, [tex]\(\sqrt[7]{x^3}\)[/tex] can be rewritten as [tex]\(x^{3/7}\)[/tex].
