Real Numbers and Algebra Essentials

Objectives

This section introduces foundational algebraic concepts essential for precalculus. Students will learn to:

• Evaluate numerical expressions

• Evaluate algebraic expressions

• Determine the domain of a variable

• Use the laws of exponents

• Evaluate square roots

Order of Operations

To correctly evaluate mathematical expressions, follow the standard order of operations:

• Parentheses: Begin with the innermost parentheses and work outward. When dividing two expressions, treat the numerator and denominator as if they are enclosed in parentheses.

• Multiplication and Division: Perform these operations from left to right.

• Addition and Subtraction: Perform these operations from left to right.

Example: Evaluating a Numerical Expression

Evaluate the expression:

Evaluating Algebraic Expressions

To evaluate an algebraic expression, substitute the given values for the variables and perform the indicated operations.

• Substitution: Replace each variable with its given value.

• Order of Operations: Follow the order of operations after substitution.

Example: Evaluating Algebraic Expressions

Given and :

Arithmetic of Quotients

When working with fractions, use the following rules:

Example: Adding, Subtracting, Multiplying, and Dividing Quotients

• (can be simplified further)

Least Common Multiple (LCM)

The least common multiple (LCM) of two numbers is the smallest positive integer that is a multiple of both numbers.

• To find the LCM, list the multiples of each number and identify the smallest common value.

Example: Finding the LCM of 10 and 15

• Multiples of 10: 10, 20, 30, 40, 50, 60, ...

• Multiples of 15: 15, 30, 45, 60, 75, ...

• The LCM is 30.

Adding Fractions Using the LCM

To add fractions with different denominators, use the LCM of the denominators as the common denominator.

• Rewrite each fraction with the LCM as the denominator.

• Add the numerators and keep the common denominator.

Example: Adding

• LCM of 10 and 15 is 30.

• Rewrite: ,

• Add:

Domain of a Variable

The domain of a variable is the set of all real numbers for which an expression is defined.

• For rational expressions, exclude values that make the denominator zero.

Example: Finding the Domain

• Expression:

• Domain: All real numbers except (since denominator cannot be zero).

Laws of Exponents

The laws of exponents are rules for simplifying expressions involving powers.

• Product of Powers:

• Power of a Power:

• Power of a Product:

• Quotient of Powers: ,

• Zero Exponent: ,

• Negative Exponent: ,

Example: Simplifying with Exponents

Principal Square Root

The principal square root of a nonnegative real number is the nonnegative number such that . It is denoted .

• For any real number ,

Example: Evaluating Square Roots

Additional info: These concepts form the basis for more advanced topics in algebra and precalculus, such as solving equations, working with functions, and manipulating algebraic expressions.