11.1 Order of Operations, Exponents & Percents
Key Takeaways
- Use PEMDAS/GEMDAS in order: grouping, exponents, multiply/divide left-to-right, add/subtract left-to-right
- a^n means a multiplied by itself n times; a^0 = 1 (a ≠ 0); negative bases in parentheses change the sign of even/odd powers
- Percent means per hundred: percent = (part / whole) × 100; convert by moving the decimal two places
- Prime numbers have exactly two distinct positive divisors (1 and themselves); 1 is not prime
- On PERT Math foundation items, rewrite the expression with clear steps so distractors from skipped order or sign errors are easy to reject
11.1 Order of Operations, Exponents & Percents
Quick Answer: Evaluate expressions with PEMDAS (Parentheses/grouping → Exponents → Multiply/Divide left-to-right → Add/Subtract left-to-right). An exponent repeats multiplication. A percent is a part of 100:
percent = (part ÷ whole) × 100. Convert among fractions, decimals, and percents by scaling by 100 or simplifying. These McCann PERT foundation skills feed every later algebra item.
PERT Mathematics opens with standard algorithms and concepts: order of operations, exponents, primes, and percents. You will not get partial credit for a “mostly correct” order—multiple-choice options are built from the most common wrong steps. This section trains the exact habits that knock those distractors out.
Order of Operations (PEMDAS / GEMDAS)
College-ready arithmetic is not left-to-right free-for-all. The agreed order is:
- Grouping — parentheses
( ), brackets[ ], braces{ }, and the bar of a fraction or radical - Exponents — powers and roots written as exponents
- Multiply and divide — same priority, left to right
- Add and subtract — same priority, left to right
GEMDAS is the same rule set with “Grouping” named first. Multiplication is not always before division; whichever appears first as you scan left to right wins.
| Priority | Operations | Critical rule |
|---|---|---|
| 1 | Grouping symbols | Innermost first; fraction bars group numerator and denominator |
| 2 | Exponents | Apply before multiply/divide |
| 3 | × and ÷ | Left to right (not “all multiplies first”) |
| 4 | + and − | Left to right |
Worked Example 1 — Mixed operations
Evaluate: 8 + 12 ÷ 4 × 3 − 2²
Step 1 — Exponents: 2² = 4, so the expression is 8 + 12 ÷ 4 × 3 − 4.
Step 2 — Multiply/divide left to right:
12 ÷ 4 = 3, then 3 × 3 = 9. Now: 8 + 9 − 4.
Step 3 — Add/subtract left to right: 8 + 9 = 17, then 17 − 4 = 13.
Answer: 13.
Trap: Doing 8 + 12 = 20 first, or treating “multiply before divide always,” yields distractors like 17, 20, or 5.
Worked Example 2 — Nested grouping and a fraction bar
Evaluate: [18 − (5 + 1)] ÷ 2 + 3 × 4
Step 1 — Innermost parentheses: 5 + 1 = 6 → [18 − 6] ÷ 2 + 3 × 4.
Step 2 — Brackets: 18 − 6 = 12 → 12 ÷ 2 + 3 × 4.
Step 3 — Multiply/divide left to right: 12 ÷ 2 = 6 and 3 × 4 = 12 → 6 + 12.
Step 4 — Add: 6 + 12 = 18.
Answer: 18.
Worked Example 3 — Absolute-value style grouping (concept)
Grouping also includes absolute-value bars when they appear: evaluate the inside first, then take the non-negative value. For |3 − 10| + 2 × 5: inside first 3 − 10 = −7, absolute value 7, then 7 + 10 = 17.
Exponents
Definition: For a positive integer n, a^n = a · a · … · a (n factors). Special cases you need cold:
| Rule | Statement | Example |
|---|---|---|
| Zero exponent | a^0 = 1 if a ≠ 0 | 7^0 = 1 |
| Power of 1 | a^1 = a | 9^1 = 9 |
| Negative base, odd power | Sign stays negative | (−2)^3 = −8 |
| Negative base, even power | Sign becomes positive | (−2)^4 = 16 |
| Missing parentheses | −2^4 = −(2^4) = −16 | Exponent applies to 2 only |
Worked Example 4 — Exponent before multiply
Evaluate: 5 · 2³ − 4²
2³ = 8 and 4² = 16 → 5 · 8 − 16 = 40 − 16 = 24.
Answer: 24.
Wrong path: (5 · 2)³ = 10³ = 1000 — never “attach” a coefficient inside a power unless parentheses say so.
Worked Example 5 — Negative bases
Compare −3² and (−3)².
−3² = −(3 · 3) = −9(square first, then apply the leading minus)(−3)² = (−3)(−3) = 9(the base is negative three)
PERT distractors love this distinction. When a calculator is available, type parentheses deliberately.
Primes and Factor Sense
A prime number is a whole number greater than 1 with exactly two distinct positive divisors: 1 and itself. The first primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, …
- 2 is the only even prime.
- 1 is not prime (only one positive divisor).
- Composite numbers have more than two positive divisors (4, 6, 8, 9, 15, …).
Prime factorization is the building block for simplifying fractions and later factoring polynomials. Example: 60 = 2² · 3 · 5.
Worked Example 6 — Is 51 prime?
Check divisibility by primes up to √51 ≈ 7.1:
51 ÷ 2 → no; ÷ 3 → 5 + 1 = 6 divisible by 3, so 51 = 3 · 17. Composite, not prime.
Percents, Decimals, and Fractions
Percent means “per hundred.” Core formulas:
percent = (part / whole) × 100part = percent/100 × wholewhole = part ÷ (percent/100)
Conversion table
| Form | To percent | To decimal | To fraction |
|---|---|---|---|
Fraction a/b | (a/b) × 100% | a ÷ b | already |
Decimal 0.d… | move decimal 2 places right + % | already | read place value |
Percent p% | already | move decimal 2 places left | p/100 simplify |
Worked Example 7 — Part–whole percent
A student answered 27 of 36 math items correctly. What percent is that?
27/36 = 0.75 → 0.75 × 100% = 75%.
Or: 27/36 = 3/4 = 75%.
Answer: 75%.
Worked Example 8 — Finding the part
What is 15% of 240?
0.15 × 240 = 36.
Answer: 36.
Worked Example 9 — Finding the whole
42 is 35% of what number?
whole = 42 ÷ 0.35.
0.35 × 120 = 42, so whole = 120.
Check: 35% of 120 = 0.35 × 120 = 42. ✓
Worked Example 10 — Fraction ↔ percent
Convert 3/8 to a percent: 3 ÷ 8 = 0.375 → 37.5%.
Convert 0.6 to a fraction: 0.6 = 6/10 = 3/5.
Convert 125% to a decimal: 1.25 (and to mixed number: 1 1/4).
Percent Increase and Decrease (foundation)
new = original × (1 ± r) where r is the rate as a decimal.
Example: A $80 textbook rises 12%. New price = 80 × 1.12 = 89.60.
Example: A score of 50 increases by 20% then falls by 20%:
50 × 1.20 = 60, then 60 × 0.80 = 48 — not back to 50. Order and successive rates matter.
PERT Strategy for Foundation Items
- Rewrite the expression on scratch paper with one operation per line.
- Circle exponents and grouping before you touch multiply/divide.
- For percents, always name part, whole, and percent so you do not invert the ratio.
- Eliminate answers that match “left-to-right ignoring PEMDAS” or “percent of wrong base.”
Mastering this section is pure free points: every linear equation and polynomial later still rests on clean arithmetic under PEMDAS.
Evaluate: 6 + 18 ÷ 3 × 2 − 5. What is the value?
Which statement is true?
A student correctly solves 21 of 28 practice questions. What percent is correct?
Which of the following numbers is prime?