In this lesson, we will learn how to simplify the product of two monomials. Our problem is to simplify the expression (2x cubed) times (4x squared).
To multiply these monomials, we first identify their components. Each monomial has a coefficient (the numerical part) and a variable part with an exponent. In 2x cubed, the coefficient is 2 and the variable is x cubed. In 4x squared, the coefficient is 4 and the variable is x squared.
When multiplying monomials, we use two key rules. First, multiply the coefficients together. Second, add the exponents of variables with the same base. This gives us the general formula: (a times x to the m) times (b times x to the n) equals (a times b) times x to the (m plus n).
Let's apply the first rule: multiply the coefficients. We have 2 times 4, which equals 8. This gives us the coefficient of our final answer.
Now we apply the second rule: add the exponents of like bases. We have x cubed times x squared. Adding the exponents 3 and 2 gives us x to the fifth power.
Now we combine our results. The coefficient is 8 and the variable part is x to the fifth power. Therefore, (2x cubed) times (4x squared) simplifies to 8x to the fifth power. This is our final answer.
Let's practice with a few more examples. First: (3x squared) times (5x to the fourth power). Multiply coefficients 3 and 5 to get 15, add exponents 2 and 4 to get 6, so the answer is 15x to the sixth power. Second: (-2x) times (6x cubed). Multiply coefficients -2 and 6 to get -12, add exponents 1 and 3 to get 4, so the answer is -12x to the fourth power.