6.2 Squares, Cubes & Optional Order of Operations

Square Numbers, Cube Numbers, and Their Roots

Square Numbers ($n^2$) and Square Roots ($\sqrt{n}$)

A square number is the product when an integer is multiplied by itself ($n \times n$). Geometrically, it represents the area of a square with side length $n$.

Key Property of Factors for Square Numbers: For positive integers, factors normally occur in distinct pairs ($a \times b$). Positive square numbers have an odd number of positive factors, because their square root pairs with itself (e.g. for $36$, $6 \times 6 = 36$, contributing only one distinct factor).

Cube Numbers ($n^3$) and Cube Roots ($\sqrt[3]{n}$)

A cube number is the product when an integer is multiplied by itself twice ($n \times n \times n$). Geometrically, it represents the volume of a cube with edge length $n$.

Special Numbers: Both Square and Cube Notice that $64$ is both a square ($8^2 = 64$) and a cube ($4^3 = 64$). Similarly, $729$ is $27^2$ and $9^3$, and $1$ is $1^2$ and $1^3$.

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Optional extension: order of operations

When a numerical expression contains multiple operations, evaluating them from left to right naively produces incorrect results. The standard mathematical hierarchy of operations is defined by the acronym BIDMAS (or BODMAS):

 ┌─────────────────────────────────────────────────────────────┐
 │ B - Brackets: Evaluate expressions inside ( ) first         │
 │ I - Indices: Evaluate powers and roots (e.g. 3², √16)       │
 │ D & M - Division and Multiplication: EQUAL PRIORITY (L to R)│
 │ A & S - Addition and Subtraction:    EQUAL PRIORITY (L to R)│
 └─────────────────────────────────────────────────────────────┘

The Two Great BIDMAS Pitfalls

Trap 1: Multiplication Does NOT Beat Division

Division and Multiplication share equal precedence. If both appear in an expression, evaluate them strictly from left to right:

$$24 \div 6 \times 2$$

• Incorrect: Evaluating multiplication first: $6 \times 2 = 12$, then $24 \div 12 = 2$.
• Correct: Moving left to right: $(24 \div 6) \times 2 = 4 \times 2 = \mathbf{8}$.

Trap 2: Addition Does NOT Beat Subtraction

Addition and Subtraction share equal precedence. Evaluate them strictly from left to right:

$$15 - 3 + 4$$

• Incorrect: Evaluating addition first: $3 + 4 = 7$, then $15 - 7 = 8$.
• Correct: Moving left to right: $(15 - 3) + 4 = 12 + 4 = \mathbf{16}$.

Step-by-Step BIDMAS Breakdown

Evaluate the expression:

$$40 - 2 \times (5 + 3^2) + 24 \div 4 \times 3$$

1. Brackets: Look inside $(5 + 3^2)$. Inside the bracket, Indices come first: $$3^2 = 9 \implies (5 + 9) = 14$$ The expression becomes: $40 - 2 \times 14 + 24 \div 4 \times 3$.
2. Division & Multiplication (Left to Right):
   • First multiplication: $2 \times 14 = 28$.
   • Next operation sequence: $24 \div 4 \times 3$.
     • Division first: $24 \div 4 = 6$.
     • Multiplication next: $6 \times 3 = 18$. The expression becomes: $40 - 28 + 18$.
3. Addition & Subtraction (Left to Right):
   • Subtraction first: $40 - 28 = 12$.
   • Addition next: $12 + 18 = \mathbf{30}$.

High-Difficulty Problem: Inserting Brackets to Make Equations True

A useful extension puzzle asks where brackets make an equation true:

• Problem: Insert one pair of brackets to make this statement true:

$$6 + 4 \times 8 - 3 = 26$$

• Without brackets: $6 + 32 - 3 = 35 \ne 26$.
• Test $(6 + 4) \times 8 - 3 = 10 \times 8 - 3 = 80 - 3 = 77 \ne 26$.
• Test $6 + 4 \times (8 - 3) = 6 + 4 \times 5 = 6 + 20 = \mathbf{26}$. Correct!

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Master Reference Summary Table

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Section Review Checklist

• Stop searching for factor pairs once you reach the square root of the number.
• Square numbers always have an ODD number of factors; all other composite numbers have an EVEN number of factors.
• The number 1 is NEITHER prime nor composite.
• 2 is the ONLY even prime number; 51, 57, 87, and 91 are composite 'imposters'.
• In BIDMAS, Division and Multiplication have equal priority; Addition and Subtraction have equal priority. Always evaluate equal-priority operations from left to right.