2.2 Rational-Number Operations & Multi-Step Order of Operations

2.2 Rational-Number Operations & Multi-Step Order of Operations

Competency 1 skill 2 is unusually specific. It names three complications by name: integer exponents, multiple levels of grouping symbols, and absolute value. Expect stems that stack all three. Skill 9 then asks you to interpret the result of such operations inside a mathematical or real-world context, so an answer of "−11" may need to be read as "an eleven-degree drop."

The hierarchy, stated precisely

+---------------------------------------------------------------------------+
|                        Order of Operations - full form                    |
+---------------------------------------------------------------------------+
|  1. Grouping    parentheses ( ), brackets [ ], braces { },                 |
|                 fraction bars, radical vinculums, absolute value | |      |
|                 -> resolve INNERMOST first, working outward               |
|  2. Exponents   and roots, right to left for stacked powers               |
|  3. Multiply / Divide   left to right, equal priority                     |
|  4. Add / Subtract      left to right, equal priority                     |
+---------------------------------------------------------------------------+

Two rules inside this list cause most errors.

Multiplication and division share a level. In 24 ÷ 6 · 2 you work left to right: 24 ÷ 6 = 4, then 4 · 2 = 8. Reading the mnemonic as "multiplication before division" gives 24 ÷ 12 = 2, which is wrong. The same applies to addition and subtraction: 10 − 3 + 5 is (10 − 3) + 5 = 12, not 10 − 8 = 2.

Grouping symbols are not only parentheses. A fraction bar groups its entire numerator and its entire denominator. In (3 + 9)/(8 − 2), both 3 + 9 and 8 − 2 must be completed before dividing, giving 12/6 = 2. Absolute value bars and radical signs behave identically.

Unary negation versus parenthesized negatives

This is the highest-yield sign rule on the competency.

• −4² means −(4²) = −(16) = −16. The exponent binds to 4, and the negative is applied afterward.
• (−4)² means (−4)(−4) = 16. The parentheses make −4 the base.

Odd powers hide the distinction: −3³ and (−3)³ both give −27, so items almost always use even exponents to expose it. When you see a negative sign in front of a power, ask whether parentheses capture the sign.

Working a stacked expression

Evaluate: 18 − 3[2³ − (7 − |4 − 9|)]² ÷ 6

1. Innermost grouping. |4 − 9| = |−5| = 5.
2. Next grouping level. 7 − 5 = 2.
3. Exponent inside brackets. 2³ = 8.
4. Complete the bracket. 8 − 2 = 6, so the expression is 18 − 3[6]² ÷ 6.
5. Exponent. 6² = 36 → 18 − 3 · 36 ÷ 6.
6. Multiply and divide, left to right. 3 · 36 = 108, then 108 ÷ 6 = 18.
7. Subtract. 18 − 18 = 0.

Step 6 is where careless work fails. Because multiplication came first reading left to right, you multiply then divide; doing 36 ÷ 6 = 6 first happens to give the same answer here, but with 3 · 36 ÷ 5 the two orders diverge.

Integer exponents, including negatives

A negative exponent produces a reciprocal, never a negative value.

The last row combines two rules: 5⁰ = 1 first, then the leading negative gives −1. And note (2/3)⁻² = (3/2)² = 9/4 — a negative exponent on a fraction flips it.

Signed-number arithmetic

Addition and subtraction. Same signs: add magnitudes, keep the sign. Different signs: subtract the smaller magnitude from the larger, keep the sign of the larger magnitude. Subtraction is addition of the opposite: a − (−b) = a + b. So −7 − (−12) = −7 + 12 = 5.

Multiplication and division. An even count of negative factors gives a positive product; an odd count gives a negative product. (−2)(−3)(−4) has three negatives, so the product is −24.

Fraction operations

• Add or subtract: find a common denominator, ideally the LCM. 5/6 − 3/8: the LCM of 6 and 8 is 24, so 20/24 − 9/24 = 11/24.
• Multiply: multiply across, cancelling common factors first. (4/9)(3/8) = (4 · 3)/(9 · 8) = 12/72 = 1/6, or cancel 4 with 8 and 3 with 9 to get (1/3)(1/2) = 1/6.
• Divide: multiply by the reciprocal. (3/5) ÷ (9/10) = (3/5)(10/9) = 30/45 = 2/3.
• Complex fractions: treat the main fraction bar as division. (2/3)/(4/9) = (2/3)(9/4) = 3/2.

Mixed numbers must be converted to improper fractions before multiplying or dividing. 2½ · 1⅓ = (5/2)(4/3) = 20/6 = 10/3 = 3⅓. Multiplying whole parts and fraction parts separately — getting 2·1 and ½·⅓ — is a classic student error and a standard distractor.

Interpreting results in context

Skill 9 asks for meaning, not just value. A stem may report that a diver descends 12.5 feet, rises 4¾ feet, then descends 6⅓ feet, and ask for the final depth relative to the surface. The arithmetic is −12.5 + 4.75 − 6.333... = −14.08..., and the interpretation is "about 14.1 feet below the surface." Answer choices will include +14.08 to catch candidates who lose the sign, and 23.58 to catch candidates who add all magnitudes.