Real Numbers and Algebra Essentials: Precalculus Study Notes

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:

$3 + [5 + 2 \cdot (9 + 7)]$ $= 3 + [5 + 2 \cdot 16]$ $= 3 + [5 + 32]$ $= 3 + 37$ $= 40$

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 $x = 2$ and $y = -3$:

• $x + 2y = 2 + 2(-3) = 2 + (-6) = -4$

• $3xy = 3 \cdot 2 \cdot (-3) = -18$

Arithmetic of Quotients

When working with fractions, use the following rules:

Operation	Formula	Conditions
Addition	$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$	$b \neq 0,\ d \neq 0$
Multiplication	$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$	$b \neq 0,\ d \neq 0$
Division	$\frac{a}{b} \div \frac{d}{c} = \frac{ad}{bc}$	$b \neq 0,\ c \neq 0,\ d \neq 0$

Example: Adding, Subtracting, Multiplying, and Dividing Quotients

• $\frac{3}{4} + \frac{1}{3} = \frac{3 \cdot 3 + 4 \cdot 1}{4 \cdot 3} = \frac{9 + 4}{12} = \frac{13}{12}$

• $\frac{2}{3} + \left(-\frac{5}{7}\right) = \frac{2 \cdot 7 + 3 \cdot (-5)}{3 \cdot 7} = \frac{14 + (-15)}{21} = \frac{-1}{21}$

• $\frac{15}{4} \div \frac{12}{5} = \frac{15}{4} \cdot \frac{5}{12} = \frac{15 \cdot 5}{4 \cdot 12} = \frac{75}{48}$ (can be simplified further)

• $\frac{2}{3} \cdot \frac{8}{5} = \frac{2 \cdot 8}{3 \cdot 5} = \frac{16}{15}$

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 $\frac{7}{10} + \frac{4}{15}$

• LCM of 10 and 15 is 30.

• Rewrite: $\frac{7}{10} = \frac{21}{30}$, $\frac{4}{15} = \frac{8}{30}$

• Add: $\frac{21}{30} + \frac{8}{30} = \frac{29}{30}$

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: $\frac{4}{x-3}$

• Domain: All real numbers except $x = 3$ (since denominator cannot be zero).

Laws of Exponents

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

• Product of Powers: $a^m \cdot a^n = a^{m+n}$

• Power of a Power: $(a^m)^n = a^{mn}$

• Power of a Product: $(ab)^n = a^n b^n$

• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$, $a \neq 0$

• Zero Exponent: $a^0 = 1$, $a \neq 0$

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$, $a \neq 0$

Example: Simplifying with Exponents

• $(x^4)^2 = x^{4 \cdot 2} = x^8$

• $x^5 \cdot x^3 = x^{5+3} = x^8$

• $\frac{x^7}{x^2} = x^{7-2} = x^5$

• $x^{-3} = \frac{1}{x^3}$

Principal Square Root

The principal square root of a nonnegative real number $a$ is the nonnegative number $b$ such that $b^2 = a$. It is denoted $\sqrt{a}$.

• For any real number $a$, $\sqrt{a^2} = |a|$

Example: Evaluating Square Roots

• $\sqrt{36} = 6$

• $\sqrt{49} = 7$

• $\sqrt{(-7)^2} = | -7 | = 7$

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.