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

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.

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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

1. Parentheses & Grouping Symbols: Evaluate expressions inside parentheses (), brackets [], braces {}, absolute value bars | |, and above/below fraction bars first.
2. Exponents & Radicals: Evaluate all powers, exponents, and radical roots from left to right.
3. Multiplication & Division: Perform multiplication and division in order of appearance from left to right. Multiplication does not automatically take precedence over division.
4. Addition & Subtraction: 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 $12 \div 3 \times 2$, division appears first from the left, so $12 \div 3 = 4$, and $4 \times 2 = 8$. Evaluating multiplication first yields $12 \div 6 = 2$, which is incorrect.

Negative Base Exponents

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

• $(-3)^2 = (-3) \times (-3) = +9$
• $-3^2 = -(3 \times 3) = -9$

Worked Example: Multi-Step Evaluation

Evaluate the following expression:

$$\text{Result} = 24 - 3 \times \left( 2^3 + 4 \right) \div 6 + |-7 + 2|$$

Step 1: Inside parentheses and absolute value

• Inside parentheses: $2^3 + 4 = 8 + 4 = 12$.
• Inside absolute value: $|-7 + 2| = |-5| = 5$.
• Expression becomes: $24 - 3 \times 12 \div 6 + 5$.

Step 2: Multiplication and Division from left to right

• First, $3 \times 12 = 36$.
• Then, $36 \div 6 = 6$.
• Expression becomes: $24 - 6 + 5$.

Step 3: Addition and Subtraction from left to right

• $24 - 6 = 18$.
• $18 + 5 = 23$.
• Final Answer: 23.

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Exponents and Exponent Laws

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

Fundamental Laws of Exponents

Worked Example: Complex Exponent Simplification

Simplify the expression leaving only positive exponents:

$$\mathcal{E} = \frac{(3x^2 y^{-3})^3}{9 x^{-4} y^2}$$

1. Apply the power of a product rule to the numerator: $$(3x^2 y^{-3})^3 = 3^3 \cdot (x^2)^3 \cdot (y^{-3})^3 = 27 x^6 y^{-9}$$
2. Write the fraction with expanded numerator: $$\mathcal{E} = \frac{27 x^6 y^{-9}}{9 x^{-4} y^2}$$
3. Simplify coefficients and apply quotient rules for variables:
   • Coefficients: $\frac{27}{9} = 3$
   • Variable $x$: $x^{6 - (-4)} = x^{6 + 4} = x^{10}$
   • Variable $y$: $y^{-9 - 2} = y^{-11}$
4. Convert negative exponent to positive exponent: $$\mathcal{E} = 3 x^{10} y^{-11} = \frac{3x^{10}}{y^{11}}$$

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Radicals and Root Operations

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

Key Radical Properties

• Product Property: $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$
• Quotient Property: $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \quad (b > 0)$
• Power-Root Cancellation: $(\sqrt[n]{a})^n = a \quad \text{and} \quad \sqrt[n]{a^n} = a \quad (\text{for } a \ge 0)$

Simplifying Radicals

To simplify a square root, factor the radicand into perfect square factors ($4, 9, 16, 25, 36, 49, 64, 81, 100, \dots$) and non-square factors.

$$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$$ $$\sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3}$$

Rationalizing Denominators

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

1. Single Term Denominator: Multiply numerator and denominator by the radical. $$\frac{5}{\sqrt{3}} = \frac{5 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{5\sqrt{3}}{3}$$
2. Binomial Denominator (Conjugates): Multiply numerator and denominator by the conjugate ($(a + \sqrt{b}) \leftrightarrow (a - \sqrt{b})$). $$\frac{4}{3 - \sqrt{5}} = \frac{4(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})} = \frac{12 + 4\sqrt{5}}{3^2 - (\sqrt{5})^2} = \frac{12 + 4\sqrt{5}}{9 - 5} = \frac{12 + 4\sqrt{5}}{4} = 3 + \sqrt{5}$$

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Scientific Notation & Real Number Properties

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

• Large Numbers: $450,000,000 = 4.5 \times 10^8$
• Small Decimals: $0.000072 = 7.2 \times 10^{-5}$

Operations in Scientific Notation

• Multiplication: Multiply coefficients and add powers of 10. $$(2.5 \times 10^4) \times (3.0 \times 10^5) = (2.5 \times 3.0) \times 10^{4+5} = 7.5 \times 10^9$$
• Division: Divide coefficients and subtract powers of 10. $$\frac{8.4 \times 10^7}{2.1 \times 10^3} = \left(\frac{8.4}{2.1}\right) \times 10^{7-3} = 4.0 \times 10^4$$

Properties of Real Numbers

• Commutative Property: $a + b = b + a$ and $a \cdot b = b \cdot a$ (Order does not change sum/product).
• Associative Property: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$ (Grouping does not change result).
• Distributive Property: $a(b + c) = ab + ac$.
• Identity Property: $a + 0 = a$ (Additive identity) and $a \cdot 1 = a$ (Multiplicative identity).
• Inverse Property: $a + (-a) = 0$ and $a \cdot \frac{1}{a} = 1 \; (a \neq 0)$.