Review of Basic Concepts: Rational Expressions, Rational Exponents, and Radical Expressions (R.5-R.7)

R.5 Rational Expressions

Definition
- A rational expression is a ratio of polynomials: $R(x) = \dfrac{P(x)}{Q(x)}$ where $Q(x) \neq 0$ .
- Treat the expression as a fraction of polynomials, with domain restrictions coming from the denominator.
Domain (Restrictions)
- Find domain by identifying values of x that make the denominator zero and excluding them.
- Process: solve $Q(x) = 0$ to obtain restricted values, then the domain is all real numbers except those values.
- Formal expression: $\text{Domain}={x\in\mathbb{R}: Q(x)\neq 0}$
- Practical note: the domain is determined from the original expression, not necessarily from any simplified form.
- Reference: R.5 Example 1 Finding the Domain (page 41).
Writing Rational Expressions in Lowest Terms
- Factor the numerator and denominator.
- Cancel common factors to obtain the lowest terms.
- Important nuance: cancelling factors may remove restrictions; always consider the original domain (do not assume the reduced form has the same domain).
- Methods mentioned in slides:
- (a) Factor; divide out the common factor. (R.5 Example 2(a), page 42)
- (b) Factor; multiply numerator and denominator by (-1); divide out the common factor. (R.5 Example 2(b), page 42)
Multiplication and Division of Rational Expressions
- Multiplication: $\dfrac{P1(x)}{Q1(x)}\cdot\dfrac{P2(x)}{Q2(x)}=\dfrac{P1(x)P2(x)}{Q1(x)Q2(x)}$ with cancellations of common factors permitted after factoring.
- Division: multiply by the reciprocal: $\dfrac{P1(x)}{Q1(x)}\div\dfrac{P2(x)}{Q2(x)}=\dfrac{P1(x)}{Q1(x)}\cdot\dfrac{Q2(x)}{P2(x)}$ then simplify by factoring and canceling common factors.
- Key step: factor completely and cancel any common factors in numerator and denominator.
- Reference: R.5 Example 3(a)–(d) (page 43) illustrating multiply/divide rational expressions with factoring and cancellation.
Addition and Subtraction of Rational Expressions
- Goal: combine fractions by finding a common denominator (LCD).
- Steps:
- Find the LCD of the denominators.
- Rewrite each term with the LCD as the denominator.
- Add/subtract the numerators.
- Simplify the resulting rational expression.
- Reference: R.5 Example 4(a)–(c) (page 44) dealing with adding and subtracting using the LCD.
Complex Fractions
- A complex fraction has a fraction in the numerator or denominator (or both).
- Simplification strategy:
- Multiply the numerator and denominator by the LCD of all fractions involved, or
- Multiply by the reciprocal of the divisor and simplify.
- Reference: R.5 Example 5(a)–(b) (page 46) showing the LCD approach for complex fractions.
Summary and Connections
- Core ideas: domain restrictions from the denominator, lowest-term form via factoring, and systematic methods for operations (multiply/divide, add/subtract, complex fractions).
- Practical tips: always check the original domain after simplification; use factoring to reveal common factors for cancellation.

R.6 Rational Exponents

Overview
- Rational exponents generalize roots and powers; key relationships include:
- $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ and more generally, for positive bases, exponent laws extend to rational exponents.
- Complex fractions revisited: the rules for exponents apply to expressions with negative or fractional exponents.
Negative Exponents
- Definition: for any nonzero $a\neq 0$ and integer $n>0$ , $a^{-n}=\dfrac{1}{a^{n}}.$ Equivalently, $a^{m/n} = \dfrac{1}{a^{-m/n}}$ when needed.
- Example form in slides: R.6 Example 1 Using the Definition of a Negative Exponent (page 50) evaluates expressions and then rewrites with no negative exponents (parts (a)–(c) and continuation (d)–(f) on pages 50–51).
The Quotient Rule and Exponent Rules
- Quotient rule: if $a\neq 0$ , then $\dfrac{a^{m}}{a^{n}}=a^{m-n}$ .
- Product rule: $a^{m}a^{n}=a^{m+n}$ .
- Power rule: $(a^{m})^{n}=a^{mn}$ .
- Combine these rules to simplify expressions with multiple bases and exponents.
- Reference: R.6 Example 2 (page 51) and Example 3 (page 51) illustrating applications of these rules (parts (a)–(d)).
Definitions for root exponents
- Definition of $a^{\frac{1}{n}}$ : $a^{1/n}=\sqrt[n]{a}$ (principal root, with domain considerations for even $n$ ).
- Definition of $a^{\frac{m}{n}}$ : $a^{m/n}=\sqrt[n]{a^{m}}$ when defined.
- Reference: R.6 Example 4 (page 52) evaluating expressions using $a^{1/n}$ and related concepts; includes cases where not all expressions are real.
Example set highlights
- Example 1 (R.6) shows evaluating expressions with negative exponents and rewriting with positive exponents.
- Example 2 (R.6) demonstrates the quotient rule with exponents.
- Example 3 (R.6) expands rules for exponents in various forms.
- Example 4 (R.6) uses the definition of $a^{1/n}$ to evaluate roots; includes real-number restrictions.
- Example 5–6 illustrate further use of $a^{m/n}$ and additional exponent manipulations.
- Example 7: factoring expressions with negative or rational exponents.
- Example 8: simplifying a fraction with negative exponents (rewrite to positive exponents, then simplify).
Important notes on real numbers and domains
- Real-valued radicals require appropriate radicands (e.g., even roots require nonnegative radicands).
- Expressions like $\sqrt[n]{x}$ may not be real for certain x when n is even; this is indicated by notes such as “not a real number.”
Summary
- Mastery of negative exponents, quotient, and product rules, and the interpretation of rational exponents as roots and powers.

R.7 Radical Expressions

Notation and concepts
- Radical expressions involve roots, denoted by $\sqrt[n]{\cdot}$ (nth roots) and principal radicals.
- Goals: simplifying radicals, performing operations with radicals, and rationalizing denominators.
Simplified radicals and radical operations
- Simplified radical: radicand simplified so no perfect-power factors remain under the radical sign (beyond obvious signs).
- Basic operations with radicals include addition/subtraction of like radicals, multiplication, and division.
Evaluating Roots and Conversions
- Example 1: Evaluating roots by converting to exponent form and computing. (R.7 Example 1, page 59)
- Write each root using exponents and evaluate: $\sqrt[n]{a} = a^{1/n}$ .
- Note that some evaluations are not real numbers depending on the radicand and index.
- Example 2: Converting from rational exponents to radicals and vice versa (R.7 Example 2, page 60).
- Write in radical form and simplify: $a^{m/n}=\sqrt[n]{a^{m}}$ ; and conversely, $\sqrt[n]{a^{m}}=a^{m/n}$ .
Converting radicals and exponents
- Example 3: Converting from radicals to rational exponents (page 60).
- Write in exponential form: $\sqrt[n]{a}=a^{1/n}$ ; then simplify.
Absolute value and radicals
- Example 4: Using absolute value to simplify roots (page 61).
- Handle expressions like $|\cdot|$ when simplifying radicals that may involve sign considerations.
- Examples 4(a)–(d); (cont.) (e)–(g) extend these ideas.
Simplifying radical expressions
- Example 5: Simplify each radical expression (page 62); (a)–(c) and continuation (d)–(f) on page 62.
- Example 6: Further simplification of radicals (page 62): parts (a)–(c) and (d)–(e).
Like radicals and combining radicals
- Example 7: Adding and subtracting like radicals (page 63): add or subtract terms with the same radical factor.
- Example 7 (cont.): Additional cases (page 63).
Multiplication and FOIL with radicals
- Example 9: Multiplying radical expressions (page 64):
- Part (a): product of radicals.
- Part (b): use FOIL and simplify.
Rationalizing denominators
- Example 10: Rationalize the denominator of a radical expression (page 65).
- Multiply by an auxiliary form to remove the radical from the denominator.
Radicals with fractions and denominators
- Example 11: Simplifying radical expressions with fractions (page 65).
- Steps include simplifying the radicand, applying the quotient rule, rationalizing the denominator, and simplifying to a common denominator if fractions are involved.
- Example 11(b): Further sequence: quotient rule, denom simplification, common denominators, and subtraction of numerators.
Rationalizing a binomial denominator
- Example 12: Rationalize a binomial denominator by multiplying numerator and denominator by the conjugate of the denominator (page 66).
Connections and practical implications
- Radical expressions are closely connected to exponent rules via the identity $a^{1/n} = \sqrt[n]{a}$ .
- Rationalizing denominators eliminates radical terms in the denominator to produce a safer, simpler form for further operations or exact values.
Summary
- Mastery of radical notations, simplification techniques, operations with radicals, and rationalization strategies for different types of denominators.

Notes on structure and references
The sections above map to the slide organization in the transcript: R.5 Rational Expressions, R.6 Rational Exponents, and R.7 Radical Expressions, including the listed examples and page references (e.g., page 41–66).
Use these notes to review the key procedures: domain determination, lowest-term forms, combining like terms via LCDs, exponent rules, and radical operations including rationalizing denominators and converting between radicals and rational exponents.