Fundamental Arithmetic Operations and Order of Operations
Arithmetic operations include addition, subtraction, multiplication, and division. However, the GRE rarely tests these in isolation. Instead, questions embed them within complex scenarios that require understanding PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Why Order Matters
Following proper order of operations prevents costly errors. The expression 2 + 3 × 4 equals 14, not 20, because multiplication happens before addition. On test day, this single rule will prevent mistakes on dozens of problems.
Operations with Different Number Types
The GRE tests how operations interact with:
- Integers and whole numbers
- Fractions and decimals
- Negative numbers
With negative numbers, remember: two negatives yield a positive, while one negative and one positive yield a negative.
Important Properties to Automate
Commutative property (a + b = b + a, ab = ba) and the distributive property (a(b + c) = ab + ac) let you manipulate expressions efficiently. These shortcuts save time when solving complex problems.
Why Flashcards Help Here
Most careless errors happen with negative numbers or multi-step operations. Flashcard practice develops automatic recall, so you execute calculations without conscious thought. This reduces errors and improves speed significantly.
Exponents and Roots: Rules and Applications
Exponents represent repeated multiplication, and mastering the rules is crucial for GRE success. You'll need to apply these rules to simplify expressions without a calculator.
Core Exponent Rules
- Product rule: a^m × a^n = a^(m+n)
- Quotient rule: a^m ÷ a^n = a^(m-n)
- Power rule: (a^m)^n = a^(mn)
- Negative exponents: a^(-n) = 1/a^n
- Zero exponents: a^0 = 1 (where a ≠ 0)
For example, 2^5 × 2^(-2) ÷ 2^0 simplifies to 2^3 = 8 using these rules.
Fractional Exponents and Roots
Fractional exponents represent roots. So a^(1/2) equals the square root of a, and a^(1/3) equals the cube root. Roots are the inverse of exponents: if x^2 = 16, then x = ±4.
Memorize perfect squares up to 15 (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225) and perfect cubes up to 5 (1, 8, 27, 64, 125). These appear constantly on the GRE.
Negative Numbers and Exponents
Pay close attention to signs. (-2)^4 = 16 (even exponent gives positive), but (-2)^3 = -8 (odd exponent keeps negative). This distinction appears in multiple question types.
Building Automaticity
These concepts feel abstract at first, but flashcard-based learning helps you internalize rules through repeated exposure. You'll move these from working memory into automatic recall.
Fractions, Decimals, and Percentage Operations
Fractions, decimals, and percentages represent different ways of expressing parts of a whole. The GRE requires fluency in converting between these forms and performing operations with each.
Fraction Operations
For addition and subtraction, find a common denominator. For multiplication, multiply numerators and denominators separately: a/b × c/d = ac/bd. For division, multiply by the reciprocal: a/b ÷ c/d = ad/bc.
Decimal Calculations
Decimals follow standard arithmetic rules, but require careful attention to decimal point placement. This matters especially in multiplication and division where students commonly misplace the decimal.
Converting Between Forms
Understand these core conversions:
- 25% = 25/100 = 1/4 = 0.25
- 50% = 1/2 = 0.5
- 33.3% = 1/3 ≈ 0.333
Percentage Problems
Percentage questions come in three main types. First, finding a part of a whole: 25% of 80 = 0.25 × 80 = 20. Second, finding what percentage one number is of another. Third, determining the original amount given a percentage change.
Percentage Increase and Decrease
If a quantity increases by 20%, it becomes 120% of its original value, or 1.2 times the original. A 30% decrease means the new amount is 70% of the original, or 0.7 times the original. This appears frequently in word problems.
Real-World Application
Calculate 15% of $240 either as 0.15 × 240 = 36 (decimal method) or 3/20 × 240 = 36 (fraction method). Flashcards help you convert seamlessly between approaches.
Simplifying Expressions and Avoiding Common Errors
Simplifying expressions requires systematic application of order of operations and careful tracking of each step. Small mistakes multiply into wrong answers.
Common Sign Errors
Forgetting to distribute negative signs causes frequent mistakes. Remember: -(a + b) = -a - b, not -a + b. Similarly, a - (-b) = a + b, not a - b. Double negatives trip up many students.
Fraction Simplification Mistakes
You can only cancel common factors, not individual terms. For example, (a + b)/b cannot simplify to a, but (ab + b)/b can simplify to a + 1 by factoring out b. This distinction separates confident test-takers from those making careless errors.
Operations with Zero and One
These edge cases appear on the GRE because they reveal conceptual misunderstanding. Remember:
- Anything multiplied by zero equals zero
- Anything multiplied by one stays the same
- Division by zero is undefined
Mistakes with Radicals and Absolute Values
Incorrectly simplifying √(a² + b²) to a + b is wrong. The expression cannot be simplified further. Similarly, |a - b| ≠ a - b when a < b. These mistakes show mechanical procedure without understanding.
Building Mental Correction
Flashcards showing correct versus incorrect simplifications side by side help you internalize proper techniques. By confronting error patterns repeatedly, you develop automatic correction mechanisms that catch mistakes before they happen during the actual exam.
Practical Study Strategies for Arithmetic Mastery
Mastering GRE arithmetic requires both conceptual understanding and procedural fluency. Strategic, spaced repetition learning develops both.
Assess Your Starting Point
Begin by identifying specific areas of weakness. If negative number operations confuse you, create focused flashcard sets addressing just that concept. If fraction multiplication troubles you, dedicate separate cards to each operation type. Diagnosis prevents wasted study time.
Design Effective Flashcard Sets
Combine three card types for comprehensive learning:
- Conceptual cards: What is the product rule for exponents?
- Computational cards: Simplify 3^4 × 3^(-2)
- Application cards: If a quantity increases by 30%, what factor do you multiply by?
Build Speed with Timed Practice
Mix flashcard review with timed calculation practice to build speed alongside accuracy. Set specific goals: aim to compute basic operations (addition through division of integers up to 100) mentally in under 5 seconds per problem.
Practice Estimation
Estimate answers before calculating exactly. This skill helps you catch careless errors on test day and identify obviously wrong answer choices.
Create Consistent Study Habits
Study sessions should be frequent but brief. Twenty to thirty minutes of focused flashcard study daily surpasses sporadic cramming. Use the spacing effect by reviewing cards you're confident about less frequently while increasing frequency of challenging concepts.
Connect to Real Problems
Practice arithmetic problems embedded in complex GRE scenarios. This shows how operations function within problem-solving contexts, not in isolation. Track your progress through quiz features in flashcard apps and celebrate improvements to maintain motivation.