To write [tex]\(\sqrt[9]{c^7}\)[/tex] as an expression with a rational exponent, we need to recall that taking the [tex]\(n\)[/tex]-th root of a number can be expressed as raising that number to the power of [tex]\(\frac{1}{n}\)[/tex].
Given [tex]\(\sqrt[9]{c^7}\)[/tex]:
1. [tex]\(\sqrt[9]{c^7}\)[/tex] is the same as [tex]\((c^7)^{1/9}\)[/tex].
2. When raising a power to another power, we multiply the exponents. Hence, [tex]\((c^7)^{1/9}\)[/tex] becomes [tex]\(c^{7 \cdot \frac{1}{9}}\)[/tex].
Therefore, the exponent simplifies to [tex]\(\frac{7}{9}\)[/tex]. The expression [tex]\(\sqrt[9]{c^7}\)[/tex] can be written as [tex]\(c^{\frac{7}{9}}\)[/tex].
None of the other options match this simplified expression. The correct answer is:
[tex]\(c^{\frac{7}{9}}\)[/tex].
