We are going to simplify the algebraic expression (3x cubed)(4x squared). When multiplying algebraic expressions with the same base, we multiply the coefficients and add the exponents.
Let's break down the expression into its components. We have two coefficients: 3 and 4, shown in red. And we have two variable terms: x cubed and x squared, shown in blue.
First, we multiply the numerical coefficients together. Three times four equals twelve.
When multiplying powers with the same base, we add the exponents. The general rule is x to the power of a times x to the power of b equals x to the power of a plus b. For example, x squared times x cubed equals x to the power of two plus three, which is x to the fifth power.
Now let's apply this rule to our specific problem. We have x cubed times x squared. According to our rule, this equals x to the power of three plus two, which equals x to the fifth power.
Now we combine our results. We multiply the coefficient result, twelve, by the variable result, x to the fifth power, to get our final answer: twelve x to the fifth power. This demonstrates the complete process of multiplying coefficients and adding exponents for same bases.
Let's review the complete solution process. We start with the original expression: (3x cubed)(4x squared). Then we separate the coefficients and variable terms: 3 times 4 and x cubed times x squared. Next we compute each part: 12 and x to the fifth power. Finally we combine them to get our answer: 12x to the fifth power.
Let's practice with another example: (2x to the fourth power)(5x cubed). We apply the same method. First, multiply the coefficients: 2 times 5 equals 10. Second, add the exponents: x to the fourth power times x cubed equals x to the power of four plus three, which is x to the seventh power. Therefore, our final answer is 10x to the seventh power.