Problem 72 - Introductory and Intermediate Algebra for College Students 4th

Question

Identify the terms in each algebraic expression. $$8 a-7 a b-13$$

Step-by-Step Solution

Answer

The terms in the expression \(8a-7ab-13\) are \(8a\), \(-7ab\), and \(-13\).

Step 1: Identification of Terms

Start with the algebraic expression \(8a-7ab-13\). Looking at the expression, observe the signs (here '+', '-'), which usually indicate the boundaries between different terms.

Step 2: Term Identification

The three terms in the algebraic expression are \(8a\), \(-7ab\), and \(-13\). Note that the negative sign is included as part of the terms \(-7ab\) and \(-13\) since they are subtracted from the other term.

Key Concepts

Terms in Algebra

In algebra, understanding what terms are is essential. Terms in algebra are the building blocks of expressions. These terms can be numerical, variable, or a combination of both. Simply put, a term is a single mathematical expression. For example, in the expression \(8a - 7ab - 13\), each component separated by a '+' or '-' sign is a term.
- **Numerical terms**: these are numbers without variables, like \(13\) or \(25\).- **Variable terms**: these include variables, such as \(x\) or \(y\), and can also involve coefficients (numbers that are multiplied by the variables) like \(8a\).
Terms can also include:

• Products of numbers and variables (e.g., \(-7ab\)).
• Exponents, which are variables raised to powers.

Always keep in mind, terms are separated by '+' and '-' operators, and those signs are crucial for the term's identity.

Identifying Terms

When you come across an algebraic expression, identifying the terms is a vital step. The terms in an expression such as \(8a - 7ab - 13\) are determined by their separation points, which, in this case, are the subtraction operators.
To identify terms effectively:

• Look for '+' or '-' signs. They mark where one term ends and another begins.
• Notice that the signs in front of each term are part of the terms themselves. This is crucial when a term is subtraction-based like \(-7ab\) or \(-13\).
• Variables and coefficients stick together to form a whole term.

Breaking down the expression \(8a - 7ab - 13\):

• The first term is \(8a\), where '8' is the coefficient and 'a' is the variable.
• The second term is \(-7ab\), with '-7' as the coefficient, and 'ab' as the variables implied with multiplication.
• The third term is \(-13\), a constant term with no variable, where the negative sign highlights subtraction from the previous term.

Algebraic Operations

Algebraic operations are fundamental tools used to manipulate expressions, helping to simplify or solve equations. The primary operations include addition, subtraction, multiplication, and division. In the context of terms like those in \(8a - 7ab - 13\), these operations help us:

• Combine like terms: Terms with the same variables and exponents can be combined using addition or subtraction.
• Apply distributive properties especially when dealing with multiplication across terms.
• Simplify expressions by performing inverse operations when necessary, such as adding \(7ab\) to both sides to isolate terms.

Consider this when working through an expression. For instance, in \(8a - 7ab - 13\): - Although we can't combine these terms directly due to differing variables, understanding which operations we're using lets us know if the expression is already simplified fully. In summary, understanding how each operation affects algebraic expressions can greatly enhance your problem-solving skills.