a) \( \left(5 \cdot 10^{3}\right)^{3} \)

To solve \( \left(5 \cdot 10^{3}\right)^{3} \), you can apply the power of a product rule. This means you'll get \( 5^{3} \cdot (10^{3})^{3} \). Calculating each part, \( 5^{3} = 125 \) and \( (10^{3})^{3} = 10^{9} \), combining both gives you \( 125 \cdot 10^{9} \) or \( 1.25 \cdot 10^{11} \) in scientific notation! Did you know that powers of ten can make complex calculations a breeze? Scientists and engineers use this system to quickly express large numbers, like the distance to a star or the size of a virus, simplifying communication and computation. When you see \( 10^{-6} \), that's commonly nanotechnology—it’s all about those tiny scales!