91Ó°ÊÓ

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. They are mathematical phrases that represent relationships between quantities. In the given exercise, the algebraic expressions are the terms \(x^{4} y^{3}\) and \(x y^{2}\).

These expressions comprise variables \(x\) and \(y\) raised to specific powers. Understanding algebraic expressions allows us to perform various mathematical operations, including finding their greatest common factor (GCF).

Recognizing the structure of algebraic expressions helps simplify problems and solve equations more efficiently.

Variable manipulation involves performing operations on variables within algebraic expressions. This can include adding, subtracting, multiplying, and dividing variables, as well as factoring and simplifying expressions.

In the provided example, variable manipulation is essential in determining the greatest common factor. We identify the common variables \(x\) and \(y\) in the terms \(x^{4} y^{3}\) and \(x y^{2}\).

By focusing on the variables and their exponents, we can determine the lowest powers of each variable and combine them to find the GCF.

• Identify common variables.
• Determine the lowest power for each variable.
• Combine the variables with their lowest powers.

Exponents indicate how many times a number (the base) is multiplied by itself. They are written as a small number (the exponent) to the upper-right of the base. For instance, in \(x^{4}\), \(x\) is the base, and \(4\) is the exponent, meaning \(x \times x \times x \times x\).

In the given exercise, the terms have exponents associated with the variables \(x\) and \(y\). The challenge is to find the greatest common factor by comparing the exponents of common variables.

For variable \(x\), the exponents are 4 and 1; the lower exponent is 1, so we use \(x^1\). For variable \(y\), the exponents are 3 and 2; the lower exponent is 2, so we use \(y^2\). Thus, the GCF of the terms is \(x^1 y^2\).

Always identify the smallest exponent for each variable when finding the GCF in algebraic expressions.