We are given the algebraic expression (3x^2y)(2x^3y^2). Our goal is to simplify this product of two monomials. Let's break it down and see how to multiply these terms systematically.
Let's identify the components of our expression. We have two monomials: 3x^2y and 2x^3y^2. Each monomial consists of a coefficient and variable terms. In the first monomial, 3 is the coefficient and x^2y are the variables. In the second monomial, 2 is the coefficient and x^3y^2 are the variables. We can rewrite the product as (coefficient times variables) times (coefficient times variables).
The first step in multiplying monomials is to multiply the coefficients together. In our expression, we have coefficients 3 and 2. Multiplying them gives us 3 times 2 equals 6. Now we can write our partial result as 6 times the product of the variable terms: 6(x^2y)(x^3y^2).
Now we apply the product rule for exponents. The rule states that when multiplying terms with the same base, we add their exponents: x to the power of a times x to the power of b equals x to the power of (a plus b). Let's apply this to our x terms: x squared times x cubed equals x to the power of (2 plus 3) which is x to the fifth power. For the y terms, we have y to the first power times y squared equals y to the power of (1 plus 2) which is y cubed.
Now we combine all the parts to get our final simplified expression. Starting with our partial result 6(x^2y)(x^3y^2), we can rewrite this as 6 times x squared times x cubed times y times y squared. Applying our exponent rules, this becomes 6 times x to the power of (2 plus 3) times y to the power of (1 plus 2). Simplifying the exponents gives us 6 times x to the fifth power times y cubed. Therefore, our final simplified expression is 6x^5y^3.
Let's review the complete solution path. We started with (3x^2y)(2x^3y^2). We separated the coefficients and variables, then multiplied the coefficients: 3 times 2 equals 6. Next, we applied the exponent rules by adding exponents of like variables: x squared times x cubed became x to the fifth power, and y times y squared became y cubed. Finally, we combined all parts to get our simplified expression: 6x^5y^3. The key steps for multiplying monomials are: first, multiply coefficients; second, add exponents of like variables; and third, combine the results. This process works for any monomial multiplication problem.