a. \( \sqrt[5]{4 a^{8} b^{14} c^{5}} \)

Let's break this expression down a bit! The fifth root of \( 4 a^{8} b^{14} c^{5} \) means we are looking for how to simplify each part. The number \( 4 \) can be rewritten as \( 2^{2} \), and then take the fifth root gives us \( 2^{\frac{2}{5}} \). For the variables, using the rule that \( \sqrt[n]{x^{m}} = x^{\frac{m}{n}} \), we find \( a^{\frac{8}{5}} \), \( b^{\frac{14}{5}} \), and \( c^{1} \). So, putting it together, we have: \[ \sqrt[5]{4 a^{8} b^{14} c^{5}} = 2^{\frac{2}{5}} a^{\frac{8}{5}} b^{\frac{14}{5}} c. \] Now it looks much tidier! Some people might forget to check their fractional powers when simplifying. Remember that \( a^{\frac{8}{5}} \) can also be expressed as \( a^{1} \cdot a^{\frac{3}{5}} \), which could be useful for further simplification. It's like uncovering hidden treasures! Always double-check to see if you've missed anything when dealing with exponents!