Algebra Essentials and College Algebra Study Guide

Foundations and Etymology of Algebra

• Etymology: The word "Algebra" originates from the Arabic term al-jabr.
• Definition: Algebra is a foundational branch of mathematics that utilizes mathematical statements to describe relationships between things that vary.
• Function: It allows for basic operations to be performed without the necessity of using specific numbers.

The System of Real Numbers

• Natural Numbers ($N$): These are the numbers used for counting: $\{1, 2, 3, \dots\}$.
• Whole Numbers ($W$): This set includes all natural numbers plus zero: $\{0, 1, 2, 3, \dots\}$.
• Integers ($I$): This set adds the negative natural numbers to the set of whole numbers: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.
• Rational Numbers ($Q$): Includes fractions written in the form $\left\{\frac{m}{n} \mid m \text{ and } n \text{ are integers and } n \neq 0\right\}$. This set also includes:
  • Terminating Decimals: Decimals that end.
  • Repeating Decimals: Decimals that have a repeating pattern.
• Irrational Numbers ($Q'$): The set of numbers that are not rational. They are characterized by being non-repeating and non-terminating: $\left\{h \mid h \text{ is not a rational number}\right\}$.
• Subset Relationships: There is a specific hierarchical relationship between these sets:
  • $N \subset W \subset I \subset Q$
  • $Q'$ is a separate set from the rational numbers, but both are subsets of the Real Numbers.

Order of Operations (PEMDAS)

To ensure mathematical expressions are evaluated consistently and accurately, the PEMDAS acronym is used to define the system of operations:

• P(arentheses): Simplify any expressions within grouping symbols first.
• E(xponents): Simplify any expressions containing exponents or radicals.
• M(ultiplication) and D(ivision): Perform these operations in order from left to right.
• A(ddition) and S(ubtraction): Perform these operations in order from left to right.

Properties of Real Numbers

The following properties hold for real numbers $a$, $b$, and $c$:

• Closure Property:
  • Addition: $a + b = c$
  • Multiplication: $ab = c$
• Commutative Property:
  • Addition: $a + b = b + a$
  • Multiplication: $a \cdot b = b \cdot a$
• Associative Property:
  • Addition: $(a + b) + c = a + (b + c)$
  • Multiplication: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
• Distributive Property: $a \cdot (b + c) = a \cdot b + a \cdot c$
• Identity Property:
  • Additive Identity: There exists a unique real number, $0$, such that for any real number $a$, $a + 0 = a$.
  • Multiplicative Identity: There exists a unique real number, $1$, such that for any real number $a$, $a \cdot 1 = a$.
• Inverse Property:
  • Additive Inverse: Every real number $a$ has an opposite, denoted $-a$, such that $a + (-a) = 0$.
  • Multiplicative Inverse: Every nonzero real number $a$ has a reciprocal, denoted $\frac{1}{a}$, such that $a \cdot \frac{1}{a} = 1$.

Algebraic Terms and Expressions

• Constant: A symbol having a fixed value.
• Variable: A symbol having no fixed value.
• Term: The basic unit of an algebraic expression. It can be a number or a product of a number and one or more variables. Terms are classified as:
  • Like Terms: Terms that have the exact same variables raised to identical powers.
  • Unlike Terms: Terms that have different variables or differing exponents. Unlike terms cannot be combined through addition or subtraction.
• Coefficient: Any factor composition of a term.
• Algebraic Expression: A collection of constants and variables joined by algebraic operations (addition, subtraction, multiplication, and division). It does not contain an equality sign and must make mathematical and logical sense.
• Algebraic Equation: A mathematical statement indicating that two expressions are equal.
• Formula: An equation expressing a relationship between constant and variable quantities.

Polynomials and Operations

• Polynomial: An algebraic expression consisting of single or multiple terms.
  • Monomial: Contains one term.
  • Binomial: Contains two terms.
  • Trinomial: Contains three terms.
• Simplifying and Evaluating:
  • Use the properties of real numbers to simplify expressions, making them easier to evaluate.
  • Addition and subtraction operate only on like terms.
  • When subtracting polynomials, the negative sign must be distributed as needed.
  • Final answers should be written with terms in decreasing order.

Rules of Exponents

For nonzero real numbers $a$ and $b$ and integers $m$ and $n$:

• Product Rule: $a^m \cdot a^n = a^{m+n}$
• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$ (where $m > n$ for natural numbers).
• Power Rule: $(a^m)^n = a^{m \cdot n}$
• Zero Exponent Rule: $a^0 = 1$ (for any nonzero real number $a$).
• Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$
• Power of a Product Rule: $(a \cdot b)^n = a^n \cdot b^n$
• Power of a Quotient Rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Scientific Notation

• Definition: A standardized way of expressing very large or very small numbers.
• Format: A number is in scientific notation if it is written as $a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer.
• Usage: Combining scientific notation with exponent rules makes calculating large or small values significantly easier than using standard notation.

Roots and Radicals

• Principal Square Root: The non-negative number that, when multiplied by itself, equals $a$. It is written as $\sqrt{a}$, where the symbol is the radical and the value inside is the radicand.
• $n$th Root: If $a$ is a real number with at least one $n$th root, then the principal $n$th root $\sqrt[n]{a}$ is the number with the same sign as $a$ that, when raised to the $n$th power, equals $a$. The value $n$ is the index of the radical.
• Rational Exponents: An alternative way to express principal $n$th roots.
  • Conversion Formula: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$
• Translating Fractional Exponents to Radicals:
  1. Determine the power from the numerator of the exponent.
  2. Determine the root from the denominator of the exponent.
  3. Use the base as the radicand, raise the radicand to the power, and use the root as the index.
  • Example: $8^{\frac{5}{3}} = (\sqrt[3]{8})^5 = 2^5 = 32$

Simplifying and Operating on Radicals

• Simplified Form Conditions:
  1. No factors are perfect $n$th powers inside the radical of index $n$.
  2. No radicals remain in the denominator of a fraction.
  3. The radical is expressed in the lowest possible index.
• Product Rule for Radicals: If $a$ and $b$ are non-negative, $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.
• Quotient Rule for Radicals: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ (where $b \neq 0$).

Examples of Radical Simplification

• Example 1: Simplify $\sqrt[3]{40}$
  • Factor 40 to find a perfect cube: $40 = 8 \times 5$.
  • $\sqrt[3]{40} = \sqrt[3]{8 \cdot 5}$.
  • Apply product rule: $\sqrt[3]{8} \cdot \sqrt[3]{5}$.
  • Result: $2\sqrt[3]{5}$.
• Example 2: Simplify $\sqrt[4]{162x^6y^9}$
  • Identify perfect fourth powers: $162 = 81 \cdot 2$, $x^6 = x^4 \cdot x^2$, $y^9 = y^8 \cdot y$.
  • $\sqrt[4]{81 \cdot 2 \cdot x^4 \cdot x^2 \cdot y^8 \cdot y} = \sqrt[4]{81x^4y^8} \cdot \sqrt[4]{2x^2y}$.
  • $(81)^{\frac{1}{4}}(x^4)^{\frac{1}{4}}(y^8)^{\frac{1}{4}} \cdot (\sqrt[4]{2x^2y})$.
  • Result: $3xy^2(\sqrt[4]{2x^2y})$.

Radical Operations and Rationalization

• Addition and Subtraction: Requires equal radicands. Simplify each radical first.
  • Example: $5\sqrt{12} + 2\sqrt{3}$
  • Rewrite: $5\sqrt{4 \cdot 3} + 2\sqrt{3} = 5 \cdot 2\sqrt{3} + 2\sqrt{3} = 10\sqrt{3} + 2\sqrt{3}$.
  • Result: $12\sqrt{3}$.
• Rationalizing the Denominator: For an expression with a single radical in the denominator:
  1. Multiply both the numerator and denominator by the radical in the denominator.
  2. Simplify the result.
  • Example: $\frac{2\sqrt{3}}{3\sqrt{10}}$
  • Multiply by $\frac{\sqrt{10}}{\sqrt{10}}$: $\frac{2\sqrt{3} \cdot \sqrt{10}}{3\sqrt{10} \cdot \sqrt{10}} = \frac{2\sqrt{30}}{3 \cdot 10} = \frac{2\sqrt{30}}{30}$.
  • Result: $\frac{\sqrt{30}}{15}$.