PERT

The PERT Mathematics subtest contains 30 multiple-choice questions: 25 scored items and 5 unscored field-test items that look identical to the rest. The test is computer-adaptive: each answer determines the difficulty of the next question, so the path through the test differs for every student. There is no time limit. Scores range from 50 to 150, and a score of 114 is college-ready for Intermediate Algebra; many colleges require a higher score, commonly 123, for placement into College Algebra. Content runs from arithmetic with integers and fractions through linear equations, systems, polynomials, factoring, quadratics, and basic functions.

The PERT calculator. Personal calculators are not allowed. An on-screen four-function calculator (add, subtract, multiply, divide) appears only for certain questions; most items must be worked by hand. The on-screen calculator performs operations in the order they are entered, not by order of operations. Typing 3 + 4 × 2 returns 14, not 11, so apply the order of operations yourself and enter one operation at a time.

Number Types

Type	Defi nition	Examples
Wholenumbers	Zeroandallpositiveintegers	0 1 2 3 , , , , …
Integers	Wholenumbersandtheirnegatives	−3 −2 −1 0 1 2 3 …, , , , , , , , …
Rational	Anyvalueexpressibleasafractionoftwointegerswitha	3/4 −5 0 .75 0 .333 , , , …
	nonzerodenominator;includesterminatingandrepeating
	decimals
Irrational	Cannotbeexpressedasasimplefraction;thedecimalnever	π √2 ,
	endsandneverrepeats
Real	Allrationalandirrationalnumberstogether	Everypointonthenumberline

Operations with Integers

Operation	Rule	Examples
Add samesigns ,	Addtheabsolutevaluesandkeepthecommonsign	( −4)+( −6) =−10
Add differentsigns ,	Subtractthesmallerabsolutevaluefromthelarger;use	−8+5=−3(8−5=3;thenegative
	thesignofthenumberwiththelargerabsolutevalue	dominates)
Subtract Multiplyordivide	Addtheopposite Same signs positive; mixed signs negative → →	1 7 − ( −4) = 7 + 4 = 11; −3 − 5 = −3 + ( −5) = −8 ( −6)( −4) =24;( −6)(4) =−24;−24÷( −6) = 4

Order of Operations

➜

➜

➜

Grouping symbols, innermost

Powers and roots

Left to right

Left to right

first

3 + 4 × 2²

= 3 + 4 × 4 exponent first: 2² = 4

= 3 + 16 multiply: 4 × 4 = 16

= 19 add

2 × (3 + (4 − 1))

= 2 × (3 + 3) innermost parentheses: 4 − 1 = 3

= 2 × 6 next parentheses: 3 + 3 = 6

= 12 multiply

With nested grouping symbols, work from the innermost pair outward.

Properties of Operations

Property	Statement	Meaning
Commutative	a+b=b+a;a·b=b·a	Orderdoesnotaffectadditionormultiplication
Associative	(a+b)+c=a+(b+c);(ab)c=a(bc)	Groupingdoesnotaffectadditionormultiplication
Distributive	a(b+c)=ab+ac	Connectsmultiplicationandaddition;essentialfor
		simplifyingexpressionsandsolvingequations
Identity	a+0=a;a·1=a	Adding0ormultiplyingby1leavesanumber
		unchanged
Inverse	a + ( −a) = 0; a · (1/a) = 1 , a ≠ 0	Pairsthatcombinetoproducetheidentities0and1

Absolute Value

Absolute value is the distance from zero on the number line, so it is never negative: |−7| = 7 and |7| = 7. Because two numbers lie at any positive distance from zero, an absolute value equation has two cases: if |x| = 5, then x = 5 or x = −5. Absolute value bars act as grouping symbols: |3 − 10| = |−7| = 7, and −|−4| = −4.

Exponents and Roots

An exponent indicates repeated multiplication: 3⁴ = 3 × 3 × 3 × 3 = 81. Roots reverse exponentiation: √49 = 7 because 7² = 49, and ∛27 = 3 because 3³ = 27.

Rule	Form	Example
Product Quotient Powerofapower Powerofaproduct	am·an=am+n am÷an=am−n (am)n=amn (ab)n=anbn	x³ ·x⁴=x⁷ x⁸÷x³ =x⁵ (x²)⁵=x¹⁰ (3x)² =9x²	1

Exponentrules(Continued)

Rule	Form	Example
Powerofaquotient	(a/b)n=an/bn	(2/5)³ =8/125
Zeroexponent	a⁰=1(a≠0)	7⁰=1;( −4x)⁰=1
Negativeexponent	a−n=1/an	2−3 =1/2³ =1/8
Rationalexponent	am/n=(ⁿ√a)m=ⁿ√(am)	82/3 =∛(8²) =∛64=4

In a rational exponent, the denominator is the root and the numerator is the power. Taking the root first keeps numbers small: 82/3 = (∛8)² = 2² = 4, 163/4 = (⁴√16)³ = 2³ = 8, and 27−1/3 = 1/∛27 = 1/3.

−3² = 9 and 2−3 = −8

−3² = −(3²) = −9, but (−3)² = 9. A negative exponent makes a reciprocal, not a negative number: 2−3 = 1/8.

Simplifying Radicals

A square root is in simplest form when the radicand has no perfect-square factor other than 1. Use √(ab) = √a · √b to pull out the largest perfect square. Perfect squares to know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

Skill	Rule	Example
Simplify	Factoroutthelargestperfectsquare	√72=√(36·2) =6√2;√50=5√2
Variables	Pairupfactors(x≥0)	√(18x³) =√(9x² ·2x)=3x√(2x)
Add/subtract	Combineonlylikeradicals(sameradicand)	3√2+5√2=8√2;√12+√27=2√3+3√3=5√3
Multiply	√a · √b = √(ab) , then simplify	√6·√15=√90=3√10
Rationalize	Multiplytopandbottombytheradicalinthe	6/√3=6√3/3=2√3
	denominator

Scientific Notation

Scientific notation writes a number as a × 10n, where the coefficient satisfies 1 ≤ a < 10. Large numbers have positive exponents (45,000,000 = 4.5 × 10⁷); small numbers have negative exponents (0.00032 = 3.2 × 10−4). To multiply, multiply the coefficients and add the exponents; to divide, divide the coefficients and subtract the exponents.

(3 × 10⁴)(2 × 10³) = 6 × 10⁷

(8 × 10⁵) ÷ (2 × 10⁻²) = 4 × 10^(5 − (−2)) = 4 × 10⁷

(5 × 10³)(4 × 10²) = 20 × 10⁵ = 2 × 10⁶

If the new coefficient is 10 or more, shift the decimal left one place and add 1 to the exponent.