Study guide
6.1 Rational Numbers, Absolute Value, and Order of Operations
ACCUPLACER Suite Study Guide Exam
Track your progress as you read
Finished sections get a check mark.
1 Chapter 1: ACCUPLACER Suite Overview & Testing Mechanics
1.1 The ACCUPLACER Testing System & Computer-Adaptive Mechanics 1.2 Placement Policies, Score Scales & Institutional Cut Scores 1.3 Test-Taking Strategies, Tool Mechanics & Pacing
2 Chapter 2: Next-Generation Reading
2.1 The Reading Test Blueprint: Set-Based and Discrete Items 2.2 Information and Ideas: Close Reading & Central Themes 2.3 Summarizing and Understanding Relationships 2.4 Rhetoric, Text Structure & Author's Purpose 2.5 Analyzing Point of View and Authorial Purpose 2.6 Analyzing Arguments: Claims, Counterclaims, and Evidence 2.7 Synthesis & Paired Text Comparative Analysis 2.8 Vocabulary in Context & Figurative Language 2.9 Passage Genre, Content Area, Mode, and Text Complexity 2.10 Answer-Choice Traps and Evidence Discipline
3 Chapter 3: Next-Generation Writing: Expression of Ideas
3.1 The Writing Test Blueprint: Passage Sets and Content Distribution 3.2 Development: Proposition and Supporting Evidence 3.3 Focus: Adding, Retaining, and Deleting for Relevance 3.4 Organization: Logical Sequence and Paragraph Order 3.5 Introductions, Conclusions, and Transitional Cohesion 3.6 Effective Language Use: Precision and Word Choice 3.7 Concision: Eliminating Wordiness and Redundancy 3.8 Style, Tone, and Voice Consistency 3.9 Syntax: Sentence Combining for Rhetorical Purpose
4 Chapter 4: Next-Generation Writing: Standard English Conventions
4.1 Sentence Boundaries: Fragments, Run-ons, and Comma Splices 4.2 Subordination and Coordination 4.3 Parallel Structure 4.4 Modifier Placement: Misplaced and Dangling Modifiers 4.5 Inappropriate Shifts in Verb Tense 4.6 Inappropriate Shifts in Verb Voice and Mood 4.7 Inappropriate Shifts in Pronoun Person and Number 4.8 Subject-Verb Agreement, Pronoun Case, and Verb Forms 4.9 Possessive Determiners, Contractions, and Confusable Adverbs 4.10 Pronoun Clarity and Pronoun-Antecedent Agreement 4.11 Frequently Confused Words, Noun Agreement, and Logical Comparison 4.12 Within-Sentence Punctuation: Semicolons, Colons, Dashes, and Apostrophes 4.13 Items in a Series and Nonrestrictive Elements 4.14 End-of-Sentence Punctuation, Hyphenation, and Unnecessary Punctuation
5 Chapter 5: Next-Generation Arithmetic
5.1 Whole Number Operations, Estimation & Multi-Step Applications 5.2 Fractions, Mixed Numbers & Operations 5.3 Decimal Operations & Place-Value Conversions 5.4 Percent Calculations, Increase, Decrease, and Applications 5.5 Number Comparisons and Equivalents Across Formats
6 Chapter 6: Next-Generation Quantitative Reasoning, Algebra, and Statistics (QAS)
6.1 Rational Numbers, Absolute Value, and Order of Operations 6.2 Exponents, Radicals, and Scientific Notation 6.3 Ratio, Proportional Relationships, and Unit Conversions 6.4 Creating and Evaluating Algebraic Expressions 6.5 Linear Equations, Inequalities, and Systems of Two Equations 6.6 Linear Applications, Functions, and Graphs 6.7 Probability and Sets 6.8 Descriptive Statistics: Center, Spread, and Graphical Displays 6.9 Geometry for Pre-Algebra and Algebra 1
7 Chapter 7: Next-Generation Advanced Algebra, Quadratics & Polynomials
7.1 Advanced Linear Equations, Systems, and Coordinate Geometry 7.2 Factoring Methods for Quadratics, Cubics, and Polynomials 7.3 Quadratic Equations, Vertex Form & Parabolic Functions 7.4 Polynomial Equations, Roots, and Graphs 7.5 Radical and Rational Equations and Functions 7.6 Geometry for Algebra 1 and Algebra 2
8 Chapter 8: Functions, Exponentials, Logarithms & Trigonometry
8.1 Function Notation, Transformations, Compositions & Inverses 8.2 Domain, Range, and Interpreting Functions in Context 8.3 Exponential & Logarithmic Functions, Laws & Modeling 8.4 Right-Triangle Trigonometry, the Unit Circle, and Radian Measure 8.5 Law of Sines, Law of Cosines, and Trigonometric Equations 8.6 Choosing Your Math Test: Arithmetic, QAS, or AAF
9 Chapter 9: WritePlacer & WritePlacer ESL Essays
9.1 WritePlacer Prompt Format, Planning, and the Writing Process 9.2 The WritePlacer Scoring Rubric & 6 Core Dimensions 9.3 Reading the WritePlacer Score Scale: What Each Band Demonstrates 9.4 Essay Architecture: Thesis, Evidence Integration & Structure 9.5 WritePlacer ESL Scoring Standards & Synthesis
10 Chapter 10: ACCUPLACER ESL Placement Suite
10.1 The ESL Placement Suite: Structure, Scoring, and Score Interpretation 10.2 ESL Reading Skills & Passage Comprehension 10.3 ESL Sentence Meaning & Context Clues 10.4 ESL Language Use & Grammatical Mechanics 10.5 ESL Listening Comprehension & Spoken Contexts
6.1 Rational Numbers, Absolute Value, and Order of Operations
Key Takeaways Explain This Topic AI
The real number system organizes values into a strict hierarchy: natural numbers, whole numbers, integers, rational numbers (expressible as a/b), and irrational numbers (non-repeating, non-terminating decimals).
Order of Operations (PEMDAS/GEMS) dictates evaluating grouping symbols first (including absolute value bars, radicals, and fraction bars), followed by exponents, then multiplication and division strictly left-to-right, and finally addition and subtraction strictly left-to-right.
A negative base raised to an even exponent evaluates to a positive result when enclosed in parentheses (e.g., (-3)^2 = 9), but evaluates to a negative result without parentheses (e.g., -3^2 = -9) because exponentiation precedes the unary negation.
Laws of exponents govern product, quotient, and power operations, including negative exponents (x^(-n) = 1/x^n) and zero exponents (x^0 = 1 for x != 0).
Simplifying radicals requires extracting perfect nth power factors from the radicand (sqrt(a*b) = sqrt(a)*sqrt(b)), combining only like radicals with identical indices and radicands, and rationalizing radical denominators.
Last updated: August 2026
Rational Numbers, Order of Operations, Exponents & Radicals
Core Foundation: The Next-Generation Quantitative Reasoning, Algebra, and Statistics (QAS) assessment measures your proficiency in manipulating real numbers, applying arithmetic hierarchies, executing exponent operations, and simplifying radical structures. Precision with signed values, negative exponents, and grouping boundaries is essential for every algebraic domain tested on the ACCUPLACER.
The Real Number System Hierarchy
All numbers evaluated on the ACCUPLACER QAS assessment reside within the set of Real Numbers ( R \mathbb{R} R ) . Understanding the classification of numbers provides the mathematical foundation for evaluating algebraic constraints, domains, and arithmetic properties. +-------------------------------------------------------------------------+
- | THE REAL NUMBER SYSTEM (ℝ) |
- +-------------------------------------------------------------------------+
- | RATIONAL NUMBERS (ℚ) │ IRRATIONAL NUMBERS |
- | Any number expressible as a/b, │ Non-terminating, |
- | where a, b ∈ ℤ and b ≠ 0. │ non-repeating decimals. |
- | (e.g., -7, 0, 3/4, 0.625, 0.333...) │ (e.g., √2, √5, π, e) |
- | │ |
- | ┌─────────────────────────────────────┐ │ |
- | │ INTEGERS (ℤ) │ │ |
- | │ {..., -3, -2, -1, 0, 1, 2, 3, ...} │ │ |
- | │ ┌───────────────────────────────┐ │ │ |
- | │ │ WHOLE NUMBERS (𝕎) │ │ │ |
- | │ │ {0, 1, 2, 3, ...} │ │ │ |
- | │ │ ┌─────────────────────────┐ │ │ │ |
- | │ │ │ NATURAL NUMBERS (ℕ) │ │ │ │ |
- | │ │ │ {1, 2, 3, 4, ...} │ │ │ │ |
- | │ │ └─────────────────────────┘ │ │ │ |
- | │ └───────────────────────────────┘ │ │ |
- | └─────────────────────────────────────┘ │ |
- +-------------------------------------------------------------------------+
- Sets of Real Numbers Defined
Natural Numbers ( N \mathbb{N} N ): Counting numbers starting from positive one: { 1 , 2 , 3 , 4 , … } \{1, 2, 3, 4, \dots\} { 1 , 2 , 3 , 4 , … } .
Whole Numbers ( W \mathbb{W} W ): Natural numbers combined with zero: { 0 , 1 , 2 , 3 , … } \{0, 1, 2, 3, \dots\} { 0 , 1 , 2 , 3 , … } .
Integers ( Z \mathbb{Z} Z ): Positive and negative whole numbers along with zero: { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\} { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } .
Rational Numbers ( Q \mathbb{Q} Q ): Numbers expressible in the fractional form a b \frac{a}{b} b a , where a a a and b b b are integers and b ≠ 0 b \neq 0 b = 0 . In decimal form, rational numbers either terminate (e.g., 3 8 = 0.375 \frac{3}{8} = 0.375 8 3 = 0.375 ) or repeat infinitely in a predictable cycle (e.g., 4 11 = 0. 36 ‾ \frac{4}{11} = 0.\overline{36} 11 4 = 0. 36 ).
Irrational Numbers: Real numbers that cannot be represented as an integer ratio. Their decimal expansions are non-terminating and non-repeating (e.g., 2 ≈ 1.4142 … \sqrt{2} \approx 1.4142\dots 2 ≈ 1.4142 … , 7 ≈ 2.6457 … \sqrt{7} \approx 2.6457\dots 7 ≈ 2.6457 … , π ≈ 3.14159 … \pi \approx 3.14159\dots π ≈ 3.14159 … ).
Operations with Signed Numbers & Absolute Value
Arithmetic errors on the ACCUPLACER most commonly stem from sign mismanagement during multi-step algebraic evaluations.
Rules for Signed Arithmetic
Addition:
Same Signs: Add the absolute values and retain the shared sign (e.g., − 8 + ( − 5 ) = − 13 -8 + (-5) = -13 − 8 + ( − 5 ) = − 13 ).
Different Signs: Subtract the smaller absolute value from the larger absolute value and retain the sign of the number with the larger absolute value (e.g., − 14 + 9 = − 5 -14 + 9 = -5 − 14 + 9 = − 5 ; 18 + ( − 7 ) = 11 18 + (-7) = 11 18 + ( − 7 ) = 11 ).
Subtraction (Adding the Additive Inverse):
Convert subtraction into addition of the opposite: a − b = a + ( − b ) a - b = a + (-b) a − b = a + ( − b ) .
Handling double negatives: a − ( − b ) = a + b a - (-b) = a + b a − ( − b ) = a + b (e.g., − 7 − ( − 12 ) = − 7 + 12 = 5 -7 - (-12) = -7 + 12 = 5 − 7 − ( − 12 ) = − 7 + 12 = 5 ).
Multiplication and Division:
Same Signs (Positive × \times × Positive or Negative × \times × Negative): Result is always positive ( ( − 6 ) × ( − 4 ) = + 24 (-6) \times (-4) = +24 ( − 6 ) × ( − 4 ) = + 24 ; − 30 − 5 = + 6 \frac{-30}{-5} = +6 − 5 − 30 = + 6 ).
Different Signs (Positive × \times × Negative or Negative × \times × Positive): Result is always negative ( ( − 7 ) × 3 = − 21 (-7) \times 3 = -21 ( − 7 ) × 3 = − 21 ; 48 − 6 = − 8 \frac{48}{-6} = -8 − 6 48 = − 8 ).
Absolute Value and Absolute Value Equations
The absolute value of a real number x x x , denoted ∣ x ∣ |x| ∣ x ∣ , represents its undirected geometric distance from zero on the real number line. Because distance cannot be negative, ∣ x ∣ ≥ 0 |x| \ge 0 ∣ x ∣ ≥ 0 for all x ∈ R x \in \mathbb{R} x ∈ R . ∣ x ∣ = { x if x ≥ 0 − x if x < 0 |x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases} ∣ x ∣ = { x − x if x ≥ 0 if x < 0
When solving absolute value equations of the form ∣ a x + b ∣ = c |ax + b| = c ∣ a x + b ∣ = c :
If c < 0 c < 0 c < 0 , there is no real solution ( ∣ a x + b ∣ = − 4 |ax + b| = -4 ∣ a x + b ∣ = − 4 is impossible).
If c = 0 c = 0 c = 0 , solve the single equation a x + b = 0 ax + b = 0 a x + b = 0 .
If c > 0 c > 0 c > 0 , split into two distinct linear cases: Case 1: a x + b = c or Case 2: a x + b = − c \text{Case 1: } ax + b = c \quad \text{or} \quad \text{Case 2: } ax + b = -c Case 1: a x + b = c or Case 2: a x + b = − c
Worked Example: Solving an Absolute Value Equation
Solve for x x x : 4 ∣ 2 x − 3 ∣ − 7 = 13 4|2x - 3| - 7 = 13 4∣2 x − 3∣ − 7 = 13 .
Step 1 (Isolate the absolute value term): Add 7 to both sides: 4 ∣ 2 x − 3 ∣ = 20 4|2x - 3| = 20 4∣2 x − 3∣ = 20 . Divide both sides by 4: ∣ 2 x − 3 ∣ = 5 |2x - 3| = 5 ∣2 x − 3∣ = 5 .
Step 2 (Split into two cases):
Case 1: 2 x − 3 = 5 ⟹ 2 x = 8 ⟹ x = 4 2x - 3 = 5 \implies 2x = 8 \implies x = 4 2 x − 3 = 5 ⟹ 2 x = 8 ⟹ x = 4
Case 2: 2 x − 3 = − 5 ⟹ 2 x = − 2 ⟹ x = − 1 2x - 3 = -5 \implies 2x = -2 \implies x = -1 2 x − 3 = − 5 ⟹ 2 x = − 2 ⟹ x = − 1
Step 3 (Verify solutions):
For x = 4 x = 4 x = 4 : 4 ∣ 2 ( 4 ) − 3 ∣ − 7 = 4 ∣ 5 ∣ − 7 = 20 − 7 = 13 4|2(4) - 3| - 7 = 4|5| - 7 = 20 - 7 = 13 4∣2 ( 4 ) − 3∣ − 7 = 4∣5∣ − 7 = 20 − 7 = 13 (Valid).
For x = − 1 x = -1 x = − 1 : 4 ∣ 2 ( − 1 ) − 3 ∣ − 7 = 4 ∣ − 5 ∣ − 7 = 20 − 7 = 13 4|2(-1) - 3| - 7 = 4|-5| - 7 = 20 - 7 = 13 4∣2 ( − 1 ) − 3∣ − 7 = 4∣ − 5∣ − 7 = 20 − 7 = 13 (Valid).
The solution set is { − 1 , 4 } \{ -1, 4 \} { − 1 , 4 } .
The Order of Operations (PEMDAS / GEMS)
To evaluate complex arithmetic and algebraic expressions unambiguously, mathematics adheres to a standardized hierarchy of operations: P / G : Parentheses & Grouping Symbols [ ( ), [ ], { }, | |, √( ), fraction bars ]
- E : Exponents & Radicals (Powers, Square Roots, Indices)
- M / D : Multiplication & Division (Evaluated STRICTLY Left-to-Right)
- A / S : Addition & Subtraction (Evaluated STRICTLY Left-to-Right)
- Critical Operational Pitfalls on the ACCUPLACER
The Left-to-Right Rule for Division and Multiplication: Multiplication does not outrank division. They share equal priority and must be evaluated in order of appearance from left to right. Example: 24 ÷ 4 × 2 = 6 × 2 = 12 ( NOT 24 ÷ 8 = 3 ) \text{Example: } 24 \div 4 \times 2 = 6 \times 2 = 12 \quad (\text{NOT } 24 \div 8 = 3) Example: 24 ÷ 4 × 2 = 6 × 2 = 12 ( NOT 24 ÷ 8 = 3 )
The Left-to-Right Rule for Subtraction and Addition: Addition does not outrank subtraction. Example: 15 − 8 + 4 = 7 + 4 = 11 ( NOT 15 − 12 = 3 ) \text{Example: } 15 - 8 + 4 = 7 + 4 = 11 \quad (\text{NOT } 15 - 12 = 3) Example: 15 − 8 + 4 = 7 + 4 = 11 ( NOT 15 − 12 = 3 )
The Unary Negation vs. Base Exponent Trap:
( − 3 ) 2 = ( − 3 ) × ( − 3 ) = + 9 (-3)^2 = (-3) \times (-3) = +9 ( − 3 ) 2 = ( − 3 ) × ( − 3 ) = + 9 (The base is − 3 -3 − 3 ).
− 3 2 = − ( 3 2 ) = − ( 9 ) = − 9 -3^2 = -(3^2) = -(9) = -9 − 3 2 = − ( 3 2 ) = − ( 9 ) = − 9 (The base is 3 3 3 ; exponentiation occurs before applying the negative sign).
Fraction Bars as Implicit Grouping Symbols: In expressions such as 18 − 6 2 + 4 \frac{18 - 6}{2 + 4} 2 + 4 18 − 6 , the numerator and denominator constitute separate, enclosed grouping envelopes. Evaluate the entire numerator ( 18 − 6 = 12 18 - 6 = 12 18 − 6 = 12 ) and entire denominator ( 2 + 4 = 6 2 + 4 = 6 2 + 4 = 6 ) before executing the division: 12 6 = 2 \frac{12}{6} = 2 6 12 = 2 .
The Laws of Integer & Rational Exponents
Exponents denote repeated multiplication: x n = x ⋅ x ⋅ x … x ⏟ n times x^n = \underbrace{x \cdot x \cdot x \dots x}_{n \text{ times}} x n = n times x ⋅ x ⋅ x … x . Algebraic expressions on the QAS assessment require fluent application of the fundamental exponent laws.
Exponent Laws Reference Table
Law Name Mathematical Identity Algebraic Example Common Error to Avoid
Product Rule x a ⋅ x b = x a + b x^a \cdot x^b = x^{a+b} x a ⋅ x b = x a + b y 4 ⋅ y 5 = y 4 + 5 = y 9 y^4 \cdot y^5 = y^{4+5} = y^9 y 4 ⋅ y 5 = y 4 + 5 = y 9 Do NOT multiply exponents ( y 20 y^{20} y 20 is wrong)
Quotient Rule x a x b = x a − b \frac{x^a}{x^b} = x^{a-b} x b x a = x a − b ( x ≠ 0 x \neq 0 x = 0 ) w 8 w 3 = w 8 − 3 = w 5 \frac{w^8}{w^3} = w^{8-3} = w^5 w 3 w 8 = w 8 − 3 = w 5 Do NOT divide exponents ( w 8 / 3 w^{8/3} w 8/3 is wrong)
Power of a Power ( x a ) b = x a b (x^a)^b = x^{ab} ( x a ) b = x ab ( m 3 ) 4 = m 3 ⋅ 4 = m 12 (m^3)^4 = m^{3 \cdot 4} = m^{12} ( m 3 ) 4 = m 3 ⋅ 4 = m 12 Do NOT add exponents ( m 7 m^7 m 7 is wrong)
Power of a Product ( x y ) a = x a y a (xy)^a = x^a y^a ( x y ) a = x a y a ( 2 a 3 b ) 3 = 2 3 ( a 3 ) 3 b 3 = 8 a 9 b 3 (2a^3b)^3 = 2^3 (a^3)^3 b^3 = 8a^9b^3 ( 2 a 3 b ) 3 = 2 3 ( a 3 ) 3 b 3 = 8 a 9 b 3 Do NOT forget to raise constant coefficient to the power ( 2 a 9 b 3 2a^9b^3 2 a 9 b 3 is wrong)
Power of a Quotient ( x y ) a = x a y a \left(\frac{x}{y}\right)^a = \frac{x^a}{y^a} ( y x ) a = y a x a ( y ≠ 0 y \neq 0 y = 0 ) ( 3 x y 2 ) 2 = 9 x 2 y 4 \left(\frac{3x}{y^2}\right)^2 = \frac{9x^2}{y^4} ( y 2 3 x ) 2 = y 4 9 x 2 Ensure exponent applies to both numerator and denominator
Zero Exponent Rule x 0 = 1 x^0 = 1 x 0 = 1 ( x ≠ 0 x \neq 0 x = 0 ) ( − 5 ) 0 = 1 (-5)^0 = 1 ( − 5 ) 0 = 1 ; − 5 0 = − 1 -5^0 = -1 − 5 0 = − 1 ; ( 4 x 2 y ) 0 = 1 (4x^2y)^0 = 1 ( 4 x 2 y ) 0 = 1 0 0 0^0 0 0 is mathematically indeterminate
Negative Exponent Rule x − n = 1 x n x^{-n} = \frac{1}{x^n} x − n = x n 1 and 1 x − n = x n \frac{1}{x^{-n}} = x^n x − n 1 = x n 4 − 2 = 1 4 2 = 1 16 4^{-2} = \frac{1}{4^2} = \frac{1}{16} 4 − 2 = 4 2 1 = 16 1 ; x − 3 y − 5 = y 5 x 3 \frac{x^{-3}}{y^{-5}} = \frac{y^5}{x^3} y − 5 x − 3 = x 3 y 5 Negative exponents indicate reciprocal position, NOT a negative numerical value
Negative Fractional Inversion ( a b ) − n = ( b a ) n \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n ( b a ) − n = ( a b ) n ( 2 3 ) − 3 = ( 3 2 ) 3 = 27 8 \left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3 = \frac{27}{8} ( 3 2 ) − 3 = ( 2 3 ) 3 = 8 27 Invert base fraction and change exponent sign to positive
Rational Exponents x m / n = x m n = ( x n ) m x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m x m / n = n x m = ( n x ) m 16 3 / 4 = ( 16 4 ) 3 = 2 3 = 8 16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8 1 6 3/4 = ( 4 16 ) 3 = 2 3 = 8 Numerator is the power; denominator is the root index
Scientific Notation Operations
Scientific notation formats very large or very small real numbers into standard form: a × 10 b where 1 ≤ ∣ a ∣ < 10 and b ∈ Z a \times 10^b \quad \text{where } 1 \le |a| < 10 \text{ and } b \in \mathbb{Z} a × 1 0 b where 1 ≤ ∣ a ∣ < 10 and b ∈ Z
Arithmetic Operations in Scientific Notation
Multiplication: Multiply the decimal coefficients and add the powers of 10: ( 3.0 × 10 5 ) × ( 4.0 × 10 3 ) = ( 3.0 × 4.0 ) × 10 5 + 3 = 12.0 × 10 8 = 1.2 × 10 9 (3.0 \times 10^5) \times (4.0 \times 10^3) = (3.0 \times 4.0) \times 10^{5+3} = 12.0 \times 10^8 = 1.2 \times 10^9 ( 3.0 × 1 0 5 ) × ( 4.0 × 1 0 3 ) = ( 3.0 × 4.0 ) × 1 0 5 + 3 = 12.0 × 1 0 8 = 1.2 × 1 0 9
Division: Divide the decimal coefficients and subtract the powers of 10: 1.6 × 10 3 4.0 × 10 7 = ( 1.6 4.0 ) × 10 3 − 7 = 0.4 × 10 − 4 = 4.0 × 10 − 5 \frac{1.6 \times 10^3}{4.0 \times 10^7} = \left(\frac{1.6}{4.0}\right) \times 10^{3-7} = 0.4 \times 10^{-4} = 4.0 \times 10^{-5} 4.0 × 1 0 7 1.6 × 1 0 3 = ( 4.0 1.6 ) × 1 0 3 − 7 = 0.4 × 1 0 − 4 = 4.0 × 1 0 − 5
Addition / Subtraction: Adjust terms so that both exponents match before adding or subtracting coefficients: ( 5.2 × 10 4 ) + ( 3.1 × 10 3 ) = ( 5.2 × 10 4 ) + ( 0.31 × 10 4 ) = ( 5.2 + 0.31 ) × 10 4 = 5.51 × 10 4 (5.2 \times 10^4) + (3.1 \times 10^3) = (5.2 \times 10^4) + (0.31 \times 10^4) = (5.2 + 0.31) \times 10^4 = 5.51 \times 10^4 ( 5.2 × 1 0 4 ) + ( 3.1 × 1 0 3 ) = ( 5.2 × 1 0 4 ) + ( 0.31 × 1 0 4 ) = ( 5.2 + 0.31 ) × 1 0 4 = 5.51 × 1 0 4
Simplifying Square Roots & Radicals
The principal square root of a non-negative real number x x x , denoted x \sqrt{x} x , is the unique non-negative number y y y such that y 2 = x y^2 = x y 2 = x .
Fundamental Radical Properties
Product Property of Radicals: a ⋅ b = a ⋅ b \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} a ⋅ b = a ⋅ b (for a , b ≥ 0 a, b \ge 0 a , b ≥ 0 )
Quotient Property of Radicals: a b = a b \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} b a = b a (for a ≥ 0 , b > 0 a \ge 0, b > 0 a ≥ 0 , b > 0 )
Simplifying Radicals by Factoring Perfect Squares
To simplify N \sqrt{N} N , identify the largest perfect square factor ( 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 , 121 , 144 , … 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, \dots 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 , 121 , 144 , … ) dividing N N N : 180 = 36 ⋅ 5 = 36 ⋅ 5 = 6 5 \sqrt{180} = \sqrt{36 \cdot 5} = \sqrt{36} \cdot \sqrt{5} = 6\sqrt{5} 180 = 36 ⋅ 5 = 36 ⋅ 5 = 6 5 72 x 5 = 36 ⋅ 2 ⋅ x 4 ⋅ x = 6 x 2 2 x \sqrt{72x^5} = \sqrt{36 \cdot 2 \cdot x^4 \cdot x} = 6x^2\sqrt{2x} 72 x 5 = 36 ⋅ 2 ⋅ x 4 ⋅ x = 6 x 2 2 x
Radical Arithmetic (Addition, Subtraction & Multiplication)
Adding/Subtracting Like Radicals: Only radicals with identical indices and identical radicands can be combined: 4 18 − 50 + 3 8 = 4 ( 3 2 ) − ( 5 2 ) + 3 ( 2 2 ) = 12 2 − 5 2 + 6 2 = 13 2 4\sqrt{18} - \sqrt{50} + 3\sqrt{8} = 4(3\sqrt{2}) - (5\sqrt{2}) + 3(2\sqrt{2}) = 12\sqrt{2} - 5\sqrt{2} + 6\sqrt{2} = 13\sqrt{2} 4 18 − 50 + 3 8 = 4 ( 3 2 ) − ( 5 2 ) + 3 ( 2 2 ) = 12 2 − 5 2 + 6 2 = 13 2
Multiplying Radicals: Multiply coefficients together and radicands together: ( 2 6 ) ( 5 3 ) = 10 18 = 10 ( 3 2 ) = 30 2 (2\sqrt{6})(5\sqrt{3}) = 10\sqrt{18} = 10(3\sqrt{2}) = 30\sqrt{2} ( 2 6 ) ( 5 3 ) = 10 18 = 10 ( 3 2 ) = 30 2
Distributing Binomials with Radicals (FOIL): ( 3 + 5 ) ( 2 − 5 ) = 6 − 3 5 + 2 5 − ( 5 ) 2 = 6 − 5 − 5 = 1 − 5 (3 + \sqrt{5})(2 - \sqrt{5}) = 6 - 3\sqrt{5} + 2\sqrt{5} - (\sqrt{5})^2 = 6 - \sqrt{5} - 5 = 1 - \sqrt{5} ( 3 + 5 ) ( 2 − 5 ) = 6 − 3 5 + 2 5 − ( 5 ) 2 = 6 − 5 − 5 = 1 − 5
Rationalizing Denominators
A radical expression is not in simplest form if a radical remains in the denominator.
Single Radical Denominator: Multiply numerator and denominator by the radical: 12 6 = 12 6 ⋅ 6 6 = 12 6 6 = 2 6 \frac{12}{\sqrt{6}} = \frac{12}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{12\sqrt{6}}{6} = 2\sqrt{6} 6 12 = 6 12 ⋅ 6 6 = 6 12 6 = 2 6
Binomial Radical Denominator (Conjugate Multiplication): The conjugate of ( a + b ) (a + \sqrt{b}) ( a + b ) is ( a − b ) (a - \sqrt{b}) ( a − b ) , using ( x + y ) ( x − y ) = x 2 − y 2 (x+y)(x-y) = x^2 - y^2 ( x + y ) ( x − y ) = x 2 − y 2 : 6 3 − 3 = 6 ( 3 + 3 ) ( 3 − 3 ) ( 3 + 3 ) = 18 + 6 3 3 2 − ( 3 ) 2 = 18 + 6 3 9 − 3 = 18 + 6 3 6 = 3 + 3 \frac{6}{3 - \sqrt{3}} = \frac{6(3 + \sqrt{3})}{(3 - \sqrt{3})(3 + \sqrt{3})} = \frac{18 + 6\sqrt{3}}{3^2 - (\sqrt{3})^2} = \frac{18 + 6\sqrt{3}}{9 - 3} = \frac{18 + 6\sqrt{3}}{6} = 3 + \sqrt{3} 3 − 3 6 = ( 3 − 3 ) ( 3 + 3 ) 6 ( 3 + 3 ) = 3 2 − ( 3 ) 2 18 + 6 3 = 9 − 3 18 + 6 3 = 6 18 + 6 3 = 3 + 3