What is the simplest form of the expression? 9. \( \sqrt{2}(\sqrt[7]{2}) \) \( \begin{array}{llll}\text { a. } 512 & \text { b. } 2^{\frac{9}{14}} & \text { c. } 2^{\frac{14}{9}} & \text { d. } 2^{\frac{1}{9}}\end{array} \)

To simplify \( \sqrt{2}(\sqrt[7]{2}) \), we can express both terms as exponents of 2. First, we know that \( \sqrt{2} \) is equivalent to \( 2^{\frac{1}{2}} \), and \( \sqrt[7]{2} \) can be written as \( 2^{\frac{1}{7}} \). Now, we can combine these using the property of exponents \( a^m \cdot a^n = a^{m+n} \): \[ 2^{\frac{1}{2}} \cdot 2^{\frac{1}{7}} = 2^{\frac{1}{2} + \frac{1}{7}}. \] Next, we need a common denominator to add \( \frac{1}{2} \) and \( \frac{1}{7} \). The least common multiple of 2 and 7 is 14. Convert \( \frac{1}{2} \) to the equivalent fraction with 14 as the denominator: \[ \frac{1}{2} = \frac{7}{14}. \] Now, convert \( \frac{1}{7} \): \[ \frac{1}{7} = \frac{2}{14}. \] Now we can add: \[ \frac{7}{14} + \frac{2}{14} = \frac{9}{14}. \] Thus, the expression simplifies to: \[ 2^{\frac{1}{2} + \frac{1}{7}} = 2^{\frac{9}{14}}. \] Therefore, the simplest form of the expression \( \sqrt{2}(\sqrt[7]{2}) \) is \( \boxed{2^{\frac{9}{14}}} \).