91Ó°ÊÓ

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (addition, subtraction, multiplication, or division). For example, in \(2x + 3y - 5\) or \(4xy^2 - 7\), both expressions consist of terms that are numbers, variables, or the product of numbers and variables.

An integral part of algebraic expressions is understanding how to identify the base and the exponent in expressions with powers, as in the exercise \( (xy)^3 \). Recognizing the constituents of these expressions allows for the application of various algebraic rules, such as the distributive property when expanding expressions or the power of a product rule when simplifying monomials.

Powers of monomials can sometimes be a tricky topic. A monomial is a single term composed of a variable or a product of variables and constants raised to natural number exponents. Consider the monomial \( x^2 \), where \( x \) is the variable and 2 is the exponent. In the case of \( (xy)^3 \) from the original exercise, the raised power affects both variables \( x \) and \( y \) equally.

Understanding that when a product of variables is raised to a power, each variable in the base is raised to that power individually, is pivotal. This is depicted as \( (xy)^3 = x^3y^3 \), applying the property where \( (ab)^n = a^n b^n \) for any variables \( a \) and \( b \) and any natural number exponent \( n \). This property is essential for simplifying algebraic expressions involving powers of monomials.

When variables are multiplied, a few rules come into play. Firstly, only like variables can be combined by addition or subtraction. However, any variables can be multiplied together, regardless of whether they are the same or different.

For instance, given two variables \( x \) and \( y \), we can multiply them to get \( xy \). This is exactly what we see in the base of the expression \( (xy)^3 \). When multiplying variables with exponents, like \( x^3y^3 \), if the variables are the same, we simply add the exponents. If the variables are different, as in \( xy \), they are presented alongside each other, often alphabetically ordered for easier readability.