In this lesson, we'll learn how to simplify the product of two monomials. Our expression is (4x squared) times (3x to the power of 1). Let's break this down and see how to simplify it step by step.
First, let's identify the components of each monomial. In 4x squared, the coefficient is 4 and the variable part is x squared. In 3x to the power of 1, the coefficient is 3 and the variable part is x to the power of 1. We'll handle the coefficients and variables separately when multiplying.
When multiplying monomials, we use two key rules. First, multiply the coefficients together. Second, add the exponents of variables with the same base. So for our general form (a times x to the m) times (b times x to the n), we get (a times b) times x to the (m plus n). Applying this to our example: multiply coefficients 4 and 3 to get 12, and add exponents 2 and 1 to get 3.
Now let's apply the first rule: multiply the coefficients. In our expression (4x squared)(3x to the power of 1), we identify the coefficients 4 and 3. Multiplying them together: 4 times 3 equals 12. So our partial result is 12 times (x squared)(x to the power of 1). Next, we'll handle the variable parts.
Now for the second rule: add the exponents of variables with the same base. In our partial result 12 times (x squared)(x to the power of 1), we identify the exponents 2 and 1. Adding them together: 2 plus 1 equals 3. Therefore, our final result is 12x cubed. This makes sense because when we multiply powers with the same base, we add their exponents.
Let's review the complete simplification process. Starting with (4x squared)(3x to the power of 1), we first separate coefficients and variables. Then we multiply the coefficients: 4 times 3 equals 12. Next we add the exponents: 2 plus 1 equals 3. This gives us our final answer: 12x cubed. To verify, we check that our coefficient is indeed 4 times 3 equals 12, and our exponent is 2 plus 1 equals 3. Both checks confirm our answer is correct.
Let's practice with a few more examples to solidify our understanding. Example 1: (2x cubed)(5x squared). Multiply coefficients 2 and 5 to get 10, add exponents 3 and 2 to get 5, so the answer is 10x to the fifth. Example 2: (6x to the fourth)(x to the first). Remember x to the first is just x, so we have coefficient 6 times 1 equals 6, and exponents 4 plus 1 equals 5, giving us 6x to the fifth. Example 3: (7x squared)(2x cubed). Coefficients 7 times 2 equals 14, exponents 2 plus 3 equals 5, so we get 14x to the fifth. In all cases, we apply the same two rules: multiply coefficients and add exponents.