(a) \( \sqrt{(a-b)^{2}} \) (b) \( \sqrt{a^{2}}+\sqrt[1]{a^{3}} \)

To simplify \( \sqrt{(a-b)^{2}} \), we remember that the square root of a square returns the absolute value of the expression inside. Thus, \( \sqrt{(a-b)^{2}} = |a-b| \). This expression tells us the distance between \( a \) and \( b \) on a number line, which is quite handy in geometry and calculus! For \( \sqrt{a^{2}}+\sqrt[1]{a^{3}} \), we simplify \( \sqrt{a^{2}} \) to \( |a| \) and \( \sqrt[1]{a^{3}} = a^{3} \). Therefore, the expression simplifies to \( |a| + a^{3} \). This gives you a fun way to mix absolute values with polynomial expressions, useful in analyzing continuous functions or when dealing with real-world scenarios like area or volume calculations!