Question

this module is intended to help you understand fractional exponents. rewrite the expression below as 9 to a single power: \\((9^5)^7 = 9^{35}\\) \\((9^8)^4 = 9^{32}\\) \\((9^{\frac{1}{9}})^9 = 9\\) let \\(x = 9^{\frac{1}{9}}\\) \\(x^9 = 9\\) solve for a positive value of \\(x\\), expressing your answer with a radical symbol. \\(x = \square\\)

Explanation:

Step1: Recall the definition of roots

To solve \(x^9 = 9\) for \(x\) (positive), we use the property that if \(y^n = a\), then \(y=\sqrt[n]{a}\) (for positive \(y\) when \(n\) is a positive integer).

Step2: Apply the root property

Given \(x^9 = 9\), by the \(n\)-th root definition, where \(n = 9\) and \(a = 9\), we get \(x=\sqrt[9]{9}\).

Answer:

\(\sqrt[9]{9}\)