6.1 Pre-Algebra & Number Operations: Exponents, Radicals & Order of Operations

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1 Chapter 1: Guide Scope, AFQT & Service-Specific Policy

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2 Chapter 2: General Science (GS)

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3 Chapter 3: Arithmetic Reasoning (AR)

3.1 Word Problems: Rates, Ratios, Proportions & Percentages 3.2 Financial & Distance Math: Interest, Profit/Loss, Speed & Time 3.3 Work & Mixture Problems: Joint Rates & Percentage Concentrations 3.4 Multi-Step Problem Solving & Algebraic Word Problems

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6 Chapter 6: Mathematics Knowledge (MK)

6.1 Pre-Algebra & Number Operations: Exponents, Radicals & Order of Operations 6.2 High School Algebra: Linear Equations, Inequalities & Polynomials 6.3 Geometry Fundamentals: Angles, Triangles, Quadrilaterals & Circles 6.4 Coordinate Geometry, Perimeter, Area & Volume Formulas

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10 Chapter 10: Spatial Assembly Practice (AO Skill)

10.1 Spatial Connections and Assembly Patterns

6.1 Pre-Algebra & Number Operations: Exponents, Radicals & Order of Operations

Key Takeaways Explain This Topic AI

Master the strict order of operations (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Apply standard exponent laws including product, quotient, power, negative exponent, and zero exponent rules with confidence.

Simplify radical expressions by factoring out perfect square factors and rationalizing denominators using conjugate multiplication.

Perform operations in scientific notation by combining coefficients and applying exponent rules to powers of 10.

Recognize and apply fundamental real number properties: commutative, associative, distributive, identity, and inverse rules.

Last updated: July 2026

6.1 Pre-Algebra & Number Operations: Exponents, Radicals & Order of Operations

Mathematics Knowledge (MK) in practice tests fundamental mathematical concepts, rules, and operations acquired throughout middle and high school mathematics. A thorough mastery of arithmetic foundations, algebraic manipulation, and numerical operations is vital for achieving high scores on the Armed Forces Classification Test.

Order of Operations (PEMDAS / BODMAS)

When evaluating algebraic or numerical expressions containing multiple operations, you must follow the standard order of operations to ensure a single, unambiguous result.

The PEMDAS Hierarchy

P arentheses & Grouping Symbols: Evaluate expressions inside parentheses () , brackets [] , braces {} , absolute value bars | | , and above/below fraction bars first.

E xponents & Radicals: Evaluate all powers, exponents, and radical roots from left to right.

M ultiplication & D ivision: Perform multiplication and division in order of appearance from left to right . Multiplication does not automatically take precedence over division.

A ddition & S ubtraction: Perform addition and subtraction in order of appearance from left to right .

Critical Pitfall: A common error in practice involves operations of equal priority. In 12 ÷ 3 × 2 12 \div 3 \times 2 12 ÷ 3 × 2 , division appears first from the left, so 12 ÷ 3 = 4 12 \div 3 = 4 12 ÷ 3 = 4 , and 4 × 2 = 8 4 \times 2 = 8 4 × 2 = 8 . Evaluating multiplication first yields 12 ÷ 6 = 2 12 \div 6 = 2 12 ÷ 6 = 2 , which is incorrect.

Negative Base Exponents

Pay close attention to parentheses when evaluating powers with negative bases:

• ( − 3 ) 2 = ( − 3 ) × ( − 3 ) = + 9 (-3)^2 = (-3) \times (-3) = +9 ( − 3 ) 2 = ( − 3 ) × ( − 3 ) = + 9
• − 3 2 = − ( 3 × 3 ) = − 9 -3^2 = -(3 \times 3) = -9 − 3 2 = − ( 3 × 3 ) = − 9
• Worked Example: Multi-Step Evaluation

Evaluate the following expression: Result = 24 − 3 × ( 2 3 + 4 ) ÷ 6 + ∣ − 7 + 2 ∣ \text{Result} = 24 - 3 \times \left( 2^3 + 4 \right) \div 6 + |-7 + 2| Result = 24 − 3 × ( 2 3 + 4 ) ÷ 6 + ∣ − 7 + 2∣

Step 1: Inside parentheses and absolute value

Inside parentheses: 2 3 + 4 = 8 + 4 = 12 2^3 + 4 = 8 + 4 = 12 2 3 + 4 = 8 + 4 = 12 .

Inside absolute value: ∣ − 7 + 2 ∣ = ∣ − 5 ∣ = 5 |-7 + 2| = |-5| = 5 ∣ − 7 + 2∣ = ∣ − 5∣ = 5 .

Expression becomes: 24 − 3 × 12 ÷ 6 + 5 24 - 3 \times 12 \div 6 + 5 24 − 3 × 12 ÷ 6 + 5 .

Step 2: Multiplication and Division from left to right

First, 3 × 12 = 36 3 \times 12 = 36 3 × 12 = 36 .

Then, 36 ÷ 6 = 6 36 \div 6 = 6 36 ÷ 6 = 6 .

Expression becomes: 24 − 6 + 5 24 - 6 + 5 24 − 6 + 5 .

Step 3: Addition and Subtraction from left to right

24 − 6 = 18 24 - 6 = 18 24 − 6 = 18 .

18 + 5 = 23 18 + 5 = 23 18 + 5 = 23 .

Final Answer: 23 .

Exponents and Exponent Laws

An exponent indicates how many times a base number is multiplied by itself. In b n b^n b n , b b b is the base and n n n is the exponent or power.

Fundamental Laws of Exponents

Exponent Rule General Formula Example

Product Rule a m ⋅ a n = a m + n a^m \cdot a^n = a^{m+n} a m ⋅ a n = a m + n x 4 ⋅ x 3 = x 4 + 3 = x 7 x^4 \cdot x^3 = x^{4+3} = x^7 x 4 ⋅ x 3 = x 4 + 3 = x 7

Quotient Rule a m a n = a m − n ( a ≠ 0 ) \frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0) a n a m ​ = a m − n ( a  = 0 ) y 9 y 4 = y 9 − 4 = y 5 \frac{y^9}{y^4} = y^{9-4} = y^5 y 4 y 9 ​ = y 9 − 4 = y 5

Power Rule ( a m ) n = a m ⋅ n (a^m)^n = a^{m \cdot n} ( a m ) n = a m ⋅ n ( z 3 ) 4 = z 3 ⋅ 4 = z 12 (z^3)^4 = z^{3 \cdot 4} = z^{12} ( z 3 ) 4 = z 3 ⋅ 4 = z 12

Power of a Product ( a b ) n = a n ⋅ b n (ab)^n = a^n \cdot b^n ( ab ) n = a n ⋅ b n ( 2 x ) 4 = 2 4 ⋅ x 4 = 16 x 4 (2x)^4 = 2^4 \cdot x^4 = 16x^4 ( 2 x ) 4 = 2 4 ⋅ x 4 = 16 x 4

Power of a Quotient ( a b ) n = a n b n ( b ≠ 0 ) \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0) ( b a ​ ) n = b n a n ​ ( b  = 0 ) ( x 3 ) 3 = x 3 27 \left(\frac{x}{3}\right)^3 = \frac{x^3}{27} ( 3 x ​ ) 3 = 27 x 3 ​

Zero Exponent Rule a 0 = 1 ( a ≠ 0 ) a^0 = 1 \quad (a \neq 0) a 0 = 1 ( a  = 0 ) 15 0 = 1 , ( − 7 x ) 0 = 1 15^0 = 1, \quad (-7x)^0 = 1 1 5 0 = 1 , ( − 7 x ) 0 = 1

Negative Exponent Rule a − n = 1 a n ( a ≠ 0 ) a^{-n} = \frac{1}{a^n} \quad (a \neq 0) a − n = a n 1 ​ ( a  = 0 ) x − 4 = 1 x 4 , 1 y − 3 = y 3 x^{-4} = \frac{1}{x^4}, \quad \frac{1}{y^{-3}} = y^3 x − 4 = x 4 1 ​ , y − 3 1 ​ = y 3

Fractional Exponent Rule a m / n = a m n = ( a n ) m a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m a m / n = n a m ​ = ( n a ​ ) m 8 2 / 3 = ( 8 3 ) 2 = 2 2 = 4 8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4 8 2/3 = ( 3 8 ​ ) 2 = 2 2 = 4

Worked Example: Complex Exponent Simplification

Simplify the expression leaving only positive exponents: E = ( 3 x 2 y − 3 ) 3 9 x − 4 y 2 \mathcal{E} = \frac{(3x^2 y^{-3})^3}{9 x^{-4} y^2} E = 9 x − 4 y 2 ( 3 x 2 y − 3 ) 3 ​

Apply the power of a product rule to the numerator: ( 3 x 2 y − 3 ) 3 = 3 3 ⋅ ( x 2 ) 3 ⋅ ( y − 3 ) 3 = 27 x 6 y − 9 (3x^2 y^{-3})^3 = 3^3 \cdot (x^2)^3 \cdot (y^{-3})^3 = 27 x^6 y^{-9} ( 3 x 2 y − 3 ) 3 = 3 3 ⋅ ( x 2 ) 3 ⋅ ( y − 3 ) 3 = 27 x 6 y − 9

Write the fraction with expanded numerator: E = 27 x 6 y − 9 9 x − 4 y 2 \mathcal{E} = \frac{27 x^6 y^{-9}}{9 x^{-4} y^2} E = 9 x − 4 y 2 27 x 6 y − 9 ​

Simplify coefficients and apply quotient rules for variables:

Coefficients: 27 9 = 3 \frac{27}{9} = 3 9 27 ​ = 3

Variable x x x : x 6 − ( − 4 ) = x 6 + 4 = x 10 x^{6 - (-4)} = x^{6 + 4} = x^{10} x 6 − ( − 4 ) = x 6 + 4 = x 10

Variable y y y : y − 9 − 2 = y − 11 y^{-9 - 2} = y^{-11} y − 9 − 2 = y − 11

Convert negative exponent to positive exponent: E = 3 x 10 y − 11 = 3 x 10 y 11 \mathcal{E} = 3 x^{10} y^{-11} = \frac{3x^{10}}{y^{11}} E = 3 x 10 y − 11 = y 11 3 x 10 ​

Radicals and Root Operations

A radical expression represents the principal root of a number. The expression a n \sqrt[n]{a} n a ​ has radical sign \sqrt{} ​ , index n n n , and radicand a a a .

Key Radical Properties

Product Property: a ⋅ b n = a n ⋅ b n \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} n a ⋅ b ​ = n a ​ ⋅ n b ​

Quotient Property: a b n = a n b n ( b > 0 ) \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \quad (b > 0) n b a ​ ​ = n b ​ n a ​ ​ ( b > 0 )

Power-Root Cancellation: ( a n ) n = a and a n n = a ( for a ≥ 0 ) (\sqrt[n]{a})^n = a \quad \text{and} \quad \sqrt[n]{a^n} = a \quad (\text{for } a \ge 0) ( n a ​ ) n = a and n a n ​ = a ( for a ≥ 0 )

Simplifying Radicals

To simplify a square root, factor the radicand into perfect square factors ( 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 , … 4, 9, 16, 25, 36, 49, 64, 81, 100, \dots 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 , … ) and non-square factors. 72 = 36 × 2 = 36 × 2 = 6 2 \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} 72 ​ = 36 × 2 ​ = 36 ​ × 2 ​ = 6 2 ​ 300 = 100 × 3 = 10 3 \sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3} 300 ​ = 100 × 3 ​ = 10 3 ​

Rationalizing Denominators

In standard mathematical form, fractions must not contain radical signs in the denominator.

Single Term Denominator: Multiply numerator and denominator by the radical. 5 3 = 5 ⋅ 3 3 ⋅ 3 = 5 3 3 \frac{5}{\sqrt{3}} = \frac{5 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{5\sqrt{3}}{3} 3 ​ 5 ​ = 3 ​ ⋅ 3 ​ 5 ⋅ 3 ​ ​ = 3 5 3 ​ ​

Binomial Denominator (Conjugates): Multiply numerator and denominator by the conjugate ( ( a + b ) ↔ ( a − b ) (a + \sqrt{b}) \leftrightarrow (a - \sqrt{b}) ( a + b ​ ) ↔ ( a − b ​ ) ). 4 3 − 5 = 4 ( 3 + 5 ) ( 3 − 5 ) ( 3 + 5 ) = 12 + 4 5 3 2 − ( 5 ) 2 = 12 + 4 5 9 − 5 = 12 + 4 5 4 = 3 + 5 \frac{4}{3 - \sqrt{5}} = \frac{4(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})} = \frac{12 + 4\sqrt{5}}{3^2 - (\sqrt{5})^2} = \frac{12 + 4\sqrt{5}}{9 - 5} = \frac{12 + 4\sqrt{5}}{4} = 3 + \sqrt{5} 3 − 5 ​ 4 ​ = ( 3 − 5 ​ ) ( 3 + 5 ​ ) 4 ( 3 + 5 ​ ) ​ = 3 2 − ( 5 ​ ) 2 12 + 4 5 ​ ​ = 9 − 5 12 + 4 5 ​ ​ = 4 12 + 4 5 ​ ​ = 3 + 5 ​

Scientific Notation & Real Number Properties

Scientific notation expresses numbers as N × 10 n N \times 10^n N × 1 0 n , where 1 ≤ ∣ N ∣ < 10 1 \le |N| < 10 1 ≤ ∣ N ∣ < 10 and n n n is an integer.

• Large Numbers: 450 , 000 , 000 = 4.5 × 10 8 450,000,000 = 4.5 \times 10^8 450 , 000 , 000 = 4.5 × 1 0 8
• Small Decimals: 0.000072 = 7.2 × 10 − 5 0.000072 = 7.2 \times 10^{-5} 0.000072 = 7.2 × 1 0 − 5
• Operations in Scientific Notation

Multiplication: Multiply coefficients and add powers of 10. ( 2.5 × 10 4 ) × ( 3.0 × 10 5 ) = ( 2.5 × 3.0 ) × 10 4 + 5 = 7.5 × 10 9 (2.5 \times 10^4) \times (3.0 \times 10^5) = (2.5 \times 3.0) \times 10^{4+5} = 7.5 \times 10^9 ( 2.5 × 1 0 4 ) × ( 3.0 × 1 0 5 ) = ( 2.5 × 3.0 ) × 1 0 4 + 5 = 7.5 × 1 0 9

Division: Divide coefficients and subtract powers of 10. 8.4 × 10 7 2.1 × 10 3 = ( 8.4 2.1 ) × 10 7 − 3 = 4.0 × 10 4 \frac{8.4 \times 10^7}{2.1 \times 10^3} = \left(\frac{8.4}{2.1}\right) \times 10^{7-3} = 4.0 \times 10^4 2.1 × 1 0 3 8.4 × 1 0 7 ​ = ( 2.1 8.4 ​ ) × 1 0 7 − 3 = 4.0 × 1 0 4

Properties of Real Numbers

Commutative Property: a + b = b + a a + b = b + a a + b = b + a and a ⋅ b = b ⋅ a a \cdot b = b \cdot a a ⋅ b = b ⋅ a (Order does not change sum/product).

Associative Property: ( a + b ) + c = a + ( b + c ) (a + b) + c = a + (b + c) ( a + b ) + c = a + ( b + c ) and ( a b ) c = a ( b c ) (ab)c = a(bc) ( ab ) c = a ( b c ) (Grouping does not change result).

Distributive Property: a ( b + c ) = a b + a c a(b + c) = ab + ac a ( b + c ) = ab + a c .

Identity Property: a + 0 = a a + 0 = a a + 0 = a (Additive identity) and a ⋅ 1 = a a \cdot 1 = a a ⋅ 1 = a (Multiplicative identity).

Inverse Property: a + ( − a ) = 0 a + (-a) = 0 a + ( − a ) = 0 and a ⋅ 1 a = 1 ( a ≠ 0 ) a \cdot \frac{1}{a} = 1 \; (a \neq 0) a ⋅ a 1 ​ = 1 ( a  = 0 ) .