Book cover for Prealgebra

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

ISBN #9781938168994

1st Edition

5,729 Questions

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210,912 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the language of algebra by defining variables, constants, expressions, and equations. It emphasizes the importance of using symbols to represent numbers, translating between algebraic notation and words, and employing the order of operations to simplify complex expressions. Mastery of these foundational skills is crucial for future topics in algebra and everyday problem-solving.

Learning Objectives

1

Use variables and algebraic symbols to represent quantities and relationships.

2

Identify and differentiate between expressions and equations.

3

Simplify expressions using exponents and the order of operations.

4

Evaluate algebraic expressions by substituting given values for variables.

Key Concepts

CONCEPT

DEFINITION

Variable

A letter or symbol that represents a number whose value can change.

Constant

A fixed number whose value does not change.

Expression

A combination of numbers, variables, and operators that represents a value; similar to a phrase in language.

Equation

Two expressions connected by an equal sign, forming a complete statement (or sentence) showing equality.

Exponent

A notation that indicates how many times a number (the base) is multiplied by itself; for example, 2鲁 means 2 multiplied by itself 3 times.

Order of Operations

A set of rules (PEMDAS: Parentheses, Exponents, Multiplication/Division from left to right, Addition/Subtraction from left to right) that determines the order in which operations are performed in an expression.

Grouping Symbols

Symbols such as parentheses (), brackets [], and braces {} used to indicate which parts of an expression should be simplified first.

Example Problems

Example 1

Translate from algebraic notation to words. $16-9$

Example 2

Translate from algebraic notation to words. $25-7$

Example 3

Translate from algebraic notation to words. 5$\cdot 6$

Example 4

Translate from algebraic notation to words. 3$\cdot 9$

Example 5

Translate from algebraic notation to words. $28 \div 4$
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Step-by-Step Explanations

QUESTION

How do you represent a variable and a constant in an algebraic expression?

STEP-BY-STEP ANSWER:

Step 1: Recognize that a variable is a symbol (like x or g) that represents a changing quantity.
Step 2: Identify the constant as a fixed number that does not change (like 3 in the expression g + 3).
Step 3: Write the expression using the variable and constant to represent the relationship (for example, if Greg's age is g, then Alex's age is represented as g + 3).
Final Answer: Variables (g) and constants (3) are used together to model relationships in algebraic expressions.

Variables and Constants

QUESTION

How do you write the product 2 路 2 路 2 in exponential form?

STEP-BY-STEP ANSWER:

Step 1: Count the number of times the base 2 is multiplied by itself; here, it appears 3 times.
Step 2: Write the multiplication in the compact exponential form as 2鲁.
Final Answer: 2 路 2 路 2 is written as 2鲁.

Exponential Notation

QUESTION

How would you simplify the expression 4 + 3 路 7 using the order of operations?

STEP-BY-STEP ANSWER:

Step 1: Identify that the expression involves addition and multiplication.
Step 2: According to PEMDAS, perform multiplication before addition; calculate 3 路 7 to get 21.
Step 3: Add the result to 4: 4 + 21 equals 25.
Final Answer: The simplified value of 4 + 3 路 7 is 25.

Order of Operations

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Common Mistakes

  • Confusing variables with constants and not recognizing that variables represent quantities that can change.
  • Failing to distinguish between expressions (incomplete phrases) and equations (complete statements with an equal sign).
  • Not following the proper order of operations, leading to incorrect simplifications.
  • Misinterpreting exponential notation by either writing it in the wrong format or improperly expanding it.