91Ó°ÊÓ

The exercise involves working with monomials. A monomial is a mathematical expression that consists of a single term. This single term can be a number, a variable, or the product of a number and one or more variables. Examples of monomials include \(-3y\), \7a\, and \4x^3\.

Monomials are very straightforward because they do not involve addition or subtraction within the term. In our exercise, \(-2x\) and \(-5x\) are both monomials because they are single terms made up of a coefficient and a variable term.

Monomials are fundamental building blocks in algebra and are used in more complex expressions and equations.

Coefficients are the numerical parts of the terms in algebraic expressions. In the expression \(-2x\), the coefficient is \(-2\), and in \(-5x\), the coefficient is \(-5\). The coefficient tells you how many times to multiply the variable.

When multiplying monomials, as seen in the exercise, the first step is to multiply the coefficients. In this case, \(-2\) and \(-5\) are the coefficients you must multiply: \(-2 \cdot -5 = 10\). Notice that the product of two negative numbers is a positive number.

Understanding how to manipulate coefficients is crucial in simplifying expressions and solving equations. Its role is to scale the variable appropriately in the context of the problem.

Variables are symbols used to represent numbers. In algebra, they are often denoted by letters such as \x, y, \text{and}\ z\. In the given exercise, \x\ is the variable component of both monomials \(-2x\) and \(-5x\).

When multiplying variables, you should add their exponents. In this case, \x\ has an implicit exponent of 1 in both terms: \x^1 \times x^1 = x^{1+1} = x^2\. This is why, when multiplying monomials, the result includes the variable raised to the sum of the original exponents.

Understanding how variables interact is crucial, as they allow algebra to generalize mathematical concepts and solve for unknown values in various contexts.