91Ó°ÊÓ

Variables are symbols used to represent numbers in mathematical expressions and equations. They are usually letters like x, y, or m. In the exercise, the variable is m. Understanding variables is crucial because they allow us to generalize mathematical concepts and apply them to different situations.

When multiplying polynomials, it’s important to keep track of the variables and their exponents (powers). In our exercise, we have two terms involving the variable m with different exponents: \(m^3\) and \(m^2\). To multiply these, we use rules for exponents, which lead to the power rule, a key concept in working with variables.

The power rule is essential when multiplying variables with exponents. It tells us how to handle the exponents when dealing with multiplication. The rule is simple: when you multiply two powers with the same base, you add the exponents. Mathematically, it’s written as: \[ m^a \times m^b = m^{a+b} \]

In the exercise, we multiply \(m^3\) and \(m^2\). Using the power rule, we add the exponents: 3 + 2 to get \(m^5\). This rule applies only when the base (the variable) is the same. So, for \(m^a \times n^b\), where m and n are different variables, the power rule doesn’t apply.

Remember, mastering the power rule makes working with polynomials much easier and less error-prone.

Constants are numbers on their own, without variables attached. They stay the same and don’t change. In the multiplication exercise, the constants are -8 and 3. These constants are essential for solving the problem, but we handle them separately from the variables.

In step 2 of the solution, we multiplied these constants together: \( -8 \times 3 = -24 \). This product becomes part of our final answer. Constants ensure that we have the correct magnitude for our final polynomial. When solving problems involving both constants and variables, always deal with the constants first before applying rules to the variables.

Understanding the role of constants helps maintain accuracy when multiplying polynomials and combining the results.