2.1 Ordering Rational Numbers, Absolute Value, Factors, Multiples & Rules of Exponents
Key Takeaways
- GED Testing Service's assessment target Q.1 covers ordering rational numbers, number properties (factors/multiples), rules of exponents, and absolute value — all tested at DOK (Depth of Knowledge) levels 1-2.
- To compare fractions and decimals of different forms, convert everything to decimals (or a common denominator) before ordering — never compare a fraction to a decimal 'by eye.'
- Rational exponents convert directly to roots: a^(1/n) means the nth root of a, and a^(m/n) means the nth root of a raised to the m power (or the nth root of a^m).
- Absolute value represents distance from zero and is always non-negative; the distance between two rational numbers a and b on the number line is |a − b|.
- These Q.1 skills anchor the test's 5 no-calculator items — GED Testing Service expects you to order numbers, apply exponent rules, and reason about absolute value without a calculator.
Why This Topic Matters
GED Testing Service's official Assessment Guide for Educators: Mathematical Reasoning opens the Quantitative Problem Solving domain with assessment target Q.1: "Apply number sense concepts, including ordering rational numbers, absolute value, multiples, factors, and exponents." This single target breaks into four graded indicators (Q.1.a through Q.1.d), and it is not a throwaway warm-up — it is the skill set the test deliberately isolates in the first 5 questions of the exam, which prohibit calculator use. Those 5 items exist specifically to confirm you can reason about number relationships, exponent rules, and absolute value mentally, before you're handed the onscreen TI-30XS Multiview calculator for the remaining 41 questions.
A rational number is any number that can be written as a ratio of two integers — this includes whole numbers, negative integers, terminating decimals (0.75), repeating decimals (0.333…), and fractions (3/4). Nearly every other Quantitative and Algebraic assessment target assumes you can already order, compare, and manipulate rational numbers fluently, so a shaky foundation here costs points across the whole test, not just on Q.1 items.
Q.1.a — Ordering Rational Numbers, Including on a Number Line
The most common trap on ordering items is comparing numbers that are written in different forms — a fraction next to a decimal next to a negative mixed number. The fix is always the same: convert everything to one common form (usually decimals, since they're easiest to compare digit-by-digit) before you order them.
Worked example: Order from least to greatest: −3.2, −7/2, −3, 3.6
- Convert −7/2 to a decimal: −7 ÷ 2 = −3.5
- Now compare all four in decimal form: −3.5, −3.2, −3, 3.6
- Remember that with negative numbers, the one further from zero is actually smaller (more negative). So −3.5 < −3.2 < −3 < 3.6
- Answer: −7/2, −3.2, −3, 3.6
This reversal of intuition — that −3.5 is less than −3.2 even though "3.5" looks bigger than "3.2" — is the single most-tested trap in this indicator. On a number line, values increase as you move right, so anything further left (more negative) is always the smaller value.
Q.1.b — Multiples, Factors, and the Distributive Property
This indicator tests whether you can apply the least common multiple (LCM), greatest common factor (GCF), or the distributive property to rewrite a numeric expression — usually to simplify a calculation or set up a common denominator.
- GCF (greatest common factor): the largest number that divides evenly into two or more numbers. Example: GCF(18, 24) = 6, because 18 = 6 × 3 and 24 = 6 × 4.
- LCM (least common multiple): the smallest number that both values divide into evenly. Example: LCM(6, 8) = 24. LCM is what you use to find a common denominator when adding fractions with unlike denominators.
- Distributive property: a(b + c) = ab + ac, used to rewrite expressions like 6 × 23 as 6 × (20 + 3) = 120 + 18 = 138 for easier mental math, or to factor an expression like 12x + 18 = 6(2x + 3) by pulling out the GCF of the coefficients.
Exam scenario: "A caterer wants to divide 18 sandwiches and 24 cookies into identical bags with no food left over, using the greatest number of bags possible. How many bags can the caterer make?" — this is a GCF problem in disguise: GCF(18, 24) = 6 bags, each with 3 sandwiches and 4 cookies.
Q.1.c — Rules of Exponents with Rational Exponents
The GED formula sheet does not list exponent rules — you must memorize them. This is a common trap: students assume every formula they'll need is printed on the sheet, but exponent rules are tested as pure number-sense recall.
| Rule | Formula | Worked Example |
|---|---|---|
| Product of powers | a^m · a^n = a^(m+n) | 2^3 · 2^2 = 2^5 = 32 |
| Quotient of powers | a^m ÷ a^n = a^(m−n) | 5^6 ÷ 5^4 = 5^2 = 25 |
| Power of a power | (a^m)^n = a^(m·n) | (3^2)^3 = 3^6 = 729 |
| Zero exponent | a^0 = 1 (a ≠ 0) | 9^0 = 1 |
| Negative exponent | a^(−n) = 1/a^n | 4^(−2) = 1/16 |
| Rational (unit fraction) exponent | a^(1/n) = the nth root of a | 8^(1/3) = 2 (cube root of 8) |
| Rational (general) exponent | a^(m/n) = (nth root of a)^m | 27^(2/3) = (cube root of 27)² = 3² = 9 |
The key conceptual leap for Q.1.c is recognizing that a rational exponent is just another way to write a root. Whenever you see a fractional exponent like x^(1/2), rewrite it as √x in your head — this converts an unfamiliar-looking expression into arithmetic you already know how to do.
Q.1.d — Absolute Value as Distance From Zero
Absolute value — written |a| — is the distance a number sits from zero on the number line, and distance is always non-negative. So |−8| = 8 and |8| = 8. The GED test extends this idea one step further: the distance between two rational numbers a and b on the number line equals |a − b| (the order doesn't matter, since |a − b| = |b − a|).
Worked example: What is the distance between −5 and 9 on the number line?
|9 − (−5)| = |9 + 5| = |14| = 14
Exam scenario: "A submarine is at an elevation of −620 feet. A helicopter is hovering at an elevation of 340 feet directly above it. What is the vertical distance between the submarine and the helicopter?" → |340 − (−620)| = |960| = 960 feet. Absolute-value distance problems on the GED are almost always dressed up as elevation, temperature, or account-balance scenarios — recognize the pattern and the arithmetic becomes routine.
Which list correctly orders the numbers from least to greatest: −3.2, −11/4, −2, 0.4?
What is the value of 16^(3/4)?
Key Takeaways
- Convert to a common form (usually decimals) before ordering rational numbers of mixed types — never trust a visual comparison between a fraction and a decimal.
- With negatives, further from zero means smaller, not larger — this reversal is the #1 tested trap for Q.1.a.
- GCF simplifies/divides into equal groups; LCM finds common denominators — know which situation calls for which.
- Memorize the exponent rules table — it is not printed on the GED formula sheet, and rational exponents always convert to a root operation.
- Absolute value = distance from zero, and the distance between two points is |a − b| — always non-negative.