a. \( 5 \sqrt[6]{x^{5}} \quad \) b. \( 5 \sqrt[3]{x^{6}} \quad \) c. \( \sqrt[6]{5 x^{5}} \quad \) d. \( 5 \sqrt[6]{x^{5}} \) 7. Write the radical expression \( \frac{8}{\sqrt[7]{x^{15}}} \) in exponential form. \( \begin{array}{llll}\text { a. } 8 x^{-\frac{7}{15}} & \text { b. } 8 x^{\frac{7}{15}} & \text { c. } 8 x^{\frac{15}{7}} & \text { d. } 8 x^{-\frac{15}{7}}\end{array} \)

To rewrite the radical expression \( \frac{8}{\sqrt[7]{x^{15}}} \) in exponential form, let's first express the radical in terms of exponents. The seventh root of \( x^{15} \) can be written as \( x^{15/7} \). Therefore, \( \sqrt[7]{x^{15}} = x^{\frac{15}{7}} \). Now, substituting that into the expression gives us: \[ \frac{8}{x^{\frac{15}{7}}} = 8 x^{-\frac{15}{7}}. \] So, the correct answer is option **d. \( 8 x^{-\frac{15}{7}} \)**. But that's not all! Did you know that using exponents over radicals can drastically simplify your calculations? It streamlines operations like multiplication and division, allowing you to manipulate numbers without getting tangled in square roots and cube roots. It’s like giving yourself a superpower when dealing with complex problems! And if you're looking to dive deeper into radical expressions and their properties, a fantastic starting point is "Algebra and Trigonometry" by Michael Sullivan. It's full of practical examples and exercises that make learning about exponents and radicals an enjoyable journey, not just a chore. Happy reading!