To rewrite the given radical expression [tex]\(\sqrt[5]{x^7}\)[/tex] as an expression with rational exponents, we can use the properties of radicals and exponents.
Here's a detailed, step-by-step solution:
1. Understand the given expression: The given expression is a fifth root of [tex]\(x\)[/tex] raised to the power of 7, which can be written as:
[tex]\[
\sqrt[5]{x^7}
\][/tex]
2. Use the radical to exponent conversion property: There is a property of exponents which states that [tex]\(\sqrt[b]{a^c}\)[/tex] can be rewritten as [tex]\(a^{\frac{c}{b}}\)[/tex]. This property allows us to convert a radical expression into a rational exponent.
3. Apply the property to the given expression: In our case, [tex]\(a\)[/tex] is [tex]\(x\)[/tex], [tex]\(c\)[/tex] is 7, and [tex]\(b\)[/tex] is 5. So, we rewrite [tex]\(\sqrt[5]{x^7}\)[/tex] as:
[tex]\[
x^{\frac{7}{5}}
\][/tex]
Thus, the radical expression [tex]\(\sqrt[5]{x^7}\)[/tex] can be rewritten as [tex]\(x^{\frac{7}{5}}\)[/tex].
Therefore, the correct answer is:
[tex]\[
x^{\frac{7}{5}}
\][/tex]
