\( \left. \begin{array} { l }{ \int 16 \sqrt[ 5 ] { x ^ { 3 } } d x } \\ { = [ ? ] \sqrt[ 5 ] { x ^ { \square } } + C } \end{array} \right. \)

To solve the integral \( \int 16 \sqrt[5]{x^3} \, dx \), we can rewrite the integrand as \( 16 x^{3/5} \). Now, using the power rule for integration, we can increase the exponent by one and divide by the new exponent: \[ \int 16 x^{3/5} \, dx = 16 \cdot \frac{x^{3/5 + 1}}{3/5 + 1} + C = 16 \cdot \frac{x^{8/5}}{8/5} + C = 16 \cdot \frac{5}{8} x^{8/5} + C = 10 x^{8/5} + C. \] So, we have: \[ = \sqrt[5]{x^{8}} + C. \] This means \( \square = 8 \). Let’s keep digging into this integral! Understanding the concept of exponents and roots can be a game-changer. You see, the joy of working with integrals is not just about solving them but also about recognizing the patterns that can simplify your work. The integration of polynomial expressions allows you to wield the power of exponents to express solutions effortlessly! And if you're eager to explore further, try different forms of \( x^{n} \) in integrals! Playing with various polynomial degrees expands your mathematical toolbox. Imagine diving headfirst into equations where your integrals have coefficients and varying exponents!