Find the least common multiple (LCM) of the expressions \(15x^2y\), \(5xy^2\), and \(30x^3y^2\).

Step 1 ：First, decompose each expression into its prime factors. \(15x^2y = 3*5*x*x*y\), \(5xy^2 = 5*x*y*y\), and \(30x^3y^2 = 2*3*5*x*x*x*y*y\).

Step 2 ：Then, for each factor, take the highest power that appears in any of the factorizations. Here, the factors are 2, 3, 5, x, and y. The highest powers are \(2^1\), \(3^1\), \(5^1\), \(x^3\), and \(y^2\).

Step 3 ：Finally, multiply these highest powers together to get the least common multiple. The LCM is \(2*3*5*x*x*x*y*y = 30x^3y^2\).