To rewrite [tex]\(\sqrt[9]{c^7}\)[/tex] as an expression with a rational exponent, follow these steps:
1. Recall the definition of a rational exponent. The nth root of a number raised to the mth power can be written as:
[tex]\[
\sqrt[n]{a^m} = a^{m/n}
\][/tex]
2. In the given expression, [tex]\(\sqrt[9]{c^7}\)[/tex], the base [tex]\(c\)[/tex] is raised to the 7th power and then the 9th root is taken. According to the rule, this can be expressed as:
[tex]\[
\sqrt[9]{c^7} = c^{7/9}
\][/tex]
3. So, the expression [tex]\(\sqrt[9]{c^7}\)[/tex] rewritten with a rational exponent is:
[tex]\[
c^{\frac{7}{9}}
\][/tex]
Among the given options, the correct choice is:
[tex]\[
c^{\frac{7}{9}}
\][/tex]
