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Chapter 2: Problem 358

In the following exercises, identify the like terms. $$ 8,8 r^{2}, 8 r, 3 r, r^{2}, 3 s $$

Short Answer

Expert verified
8, 8r^{2} and r^{2}, 8r and 3r, 3s

Step by step solution

01

- Identify constants

Look for numerical values without any variables. Here, the term is 8.
02

- Identify terms with the same variable and exponent

Find terms with the same variable raised to the same power: 1. Terms with the variable raised to the power of 2: - 8r^{2} - r^{2} 2. Terms with 'r' raised to the power of 1: - 8r - 3r 3. Single variable term raised to power 1: - 3s
03

- Group like terms

Separate the like terms identified in the previous step: 1. Constants: 8 2. Terms with \(r^{2}\): 8r^{2}, r^{2} 3. Terms with \(r\): 8r, 3r 4. Terms with \(s\): 3s

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constants in Algebra
Constants are the simplest terms in algebra. They are numbers on their own, without any attached variables. For instance, in the exercise provided, the term 8 is a constant.
A constant remains the same because it does not depend on any variable. This makes constants easy to spot when grouping like terms. Anytime you see a standalone number in an algebraic expression, you are looking at a constant.

For example:
  • In the expression 5 + 3x, the number 5 is a constant.
  • Constants can also be negative, such as -2 in the expression -2 + x.
Terms with Variables
Terms with variables are algebraic expressions that include numbers multiplied by letters (variables). These variables can represent different numbers and are often found with exponents.
In the exercise, examples of terms with variables are 8r^{2}, 8r, 3r, r^{2}, and 3s. Here, 'r' and 's' are variables and their exponents specify how many times the variable is multiplied by itself.

To fully understand terms with variables:
  • The coefficient is the number in front of the variable. In '8r', 8 is the coefficient.
  • The variable is the letter that represents an unknown value, such as 'r' or 's'.
  • The exponent indicates the power to which the variable is raised. In '8r^{2}', the exponent is 2, meaning r is squared.
Grouping Like Terms
Grouping like terms involves collecting terms in an algebraic expression that have identical variables raised to the same power. This allows us to simplify expressions and solve equations more easily.
In the exercise, the constants and terms with the same variables and exponents are grouped as follows:
  • Constants: 8
  • Terms with r^{2}: 8r^{2}, r^{2}
  • Terms with r: 8r, 3r
  • Terms with s: 3s
When grouping like terms, remember:
鈥� Combine terms that have exactly the same variable part (same variable and same exponent).
鈥� Simplify the grouped terms by adding or subtracting the coefficients.

For example, 2x + 3x becomes (2+3)x = 5x. This process helps in solving algebraic equations systematically.

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