Problem 86 - Elementary Algebra

Question

Explain why \(7 a^{2} b\) and \(5 a b^{2}\) are not like terms.

Step-by-Step Solution

Answer

The terms \(7 a^{2} b\) and \(5 a b^{2}\) are not like terms because their variables have different exponents.

Step 1: Understand the Definition of Like Terms

Like terms are terms that have the exact same variables raised to the same powers. In other words, the variables and their exponents must match exactly for the terms to be considered like terms.

Step 2: Identify the Variables and Exponents

Look at the variables and their exponents in the terms provided. For the term \(7 a^{2} b\), the variables are \(a\) and \(b\), with exponents 2 and 1, respectively. For the term \(5 a b^{2}\), the variables are also \(a\) and \(b\), but with exponents 1 and 2, respectively.

Step 3: Compare Variables and Their Exponents

Compare the variables and their exponents from both terms. In \(7 a^{2} b\), the exponents are 2 for \(a\) and 1 for \(b\). In \(5 a b^{2}\), the exponents are 1 for \(a\) and 2 for \(b\). Since the exponents for the variables do not match, the terms are not like terms.

Key Concepts

variables and exponents

In algebra, variables and exponents play key roles in forming expressions and equations. A variable is a symbol, often a letter, used to represent a number. An exponent tells us how many times the base (which can be a variable or a number) is multiplied by itself. For instance, in the term \(a^2\), \(a\) is the variable, and 2 is the exponent, indicating that \(a\) is multiplied by itself to give \(a \times a\). Similarly, in the term \(b^3\), \(b\) is the variable, and 3 is the exponent, meaning \(b \times b \times b\). Understanding the roles of variables and exponents helps in simplifying expressions and solving equations accurately.

terms in algebra

Terms in algebra are the building blocks of algebraic expressions. A term is a combination of variables, exponents, and coefficients (numbers). For example, in the term \(7a^2b\), 7 is the coefficient, \(a\) and \(b\) are the variables, and the exponents are 2 and 1, respectively. A single algebraic expression can have multiple terms separated by addition or subtraction signs. For instance, in the expression \(3x + 4y - 5z\), there are three terms: \(3x\), \(4y\), and \(5z\). Each term is treated independently when simplifying expressions or solving equations.

comparing exponents

Comparing exponents is essential when determining whether terms are like terms. Like terms are terms that have identical variables raised to the same exponents. The coefficients can differ, but the variables and their exponents must match. For example, consider the terms \(7a^2b\) and \(5ab^2\). To determine if these terms are like terms, compare their variables and exponents. In \(7a^2b\), the exponents for \(a\) and \(b\) are 2 and 1, respectively. In \(5ab^2\), the exponents for \(a\) and \(b\) are 1 and 2, respectively. Since the exponents do not match, these terms are not like terms and cannot be combined in a single expression.