We are going to learn how to simplify algebraic expressions like (2x squared) times (3x to the first power). This involves understanding how to multiply monomials, which are expressions with a single term.
Let's break down our expression into its components. In the first monomial 2x squared, we have coefficient 2 and variable x squared. In the second monomial 3x to the first, we have coefficient 3 and variable x to the first power. Understanding these parts is key to simplifying the expression.
When multiplying monomials, we first multiply the coefficients. In our example, we multiply 2 and 3 to get 6. This is the coefficient part of our final answer.
When multiplying variables with the same base, we add their exponents. So x squared times x to the first power equals x to the power of (2 plus 1), which is x cubed. This is a fundamental rule in algebra.
Now let's put it all together. We start with (2x squared)(3x to the first). First, we multiply the coefficients: 2 times 3 equals 6. Then we multiply the variables by adding exponents: x squared times x to the first equals x cubed. So our final simplified expression is 6x cubed.
Let's verify our answer by expanding it back. 6x cubed is indeed equal to (2x squared)(3x to the first). Now let's look at a couple more examples. For (4x cubed)(2x squared), we multiply coefficients 4 and 2 to get 8, and add exponents 3 and 2 to get 5, resulting in 8x to the fifth. For (5x to the first)(x to the fourth), we multiply coefficients 5 and 1 to get 5, and add exponents 1 and 4 to get 5, resulting in 5x to the fifth. The pattern is always the same: multiply coefficients and add exponents for like bases.