00-10 — BODMAS with Complex Nested Expressions

Phase: 0 — Arithmetic & Number Foundations Subject: 00-10 Prerequisites: 00-01 — Whole Number Arithmetic, 00-06 — Powers and Roots Next subject: 00-11 — Laws of Indices (Exponents)

Learning Objectives

By the end of this subject, you will be able to:

1. Apply BODMAS/PEMDAS correctly to expressions with multiple nested parentheses, including square brackets and curly braces
2. Interpret the fraction bar (vinculum) as an implicit grouping symbol and evaluate expressions written in fraction form
3. Evaluate expressions that combine exponents, roots, and all four basic operations in the correct order
4. Identify and correct common order-of-operations mistakes through systematic step-by-step reasoning
5. Translate word problems into correctly bracketed numerical expressions and evaluate them

Core Content

1. BODMAS Refresher

BODMAS (or PEMDAS) is the agreed-upon convention for the order in which we evaluate expressions:

⚠️ THIS IS CRITICAL — BODMAS is not optional. It is a universal convention that ensures everyone evaluating the same expression gets the same answer. Ignoring it does not give an "alternative interpretation" — it gives a wrong answer. Every calculator, programming language, and mathematician follows these rules.

Key refinements from 00-01: - Division and multiplication have equal precedence — work left to right - Addition and subtraction have equal precedence — work left to right - When in doubt, add brackets to make your intention clear

2. Nested Brackets — Working from the Inside Out

When brackets are nested inside other brackets, evaluate the innermost brackets first, then work outward.

Common bracket types (all equivalent — readability is the only reason for different shapes): - Parentheses: ( ) - Square brackets: [ ] - Curly braces: { }

Rule: Always resolve what's inside the innermost pair first.

Example 1: Evaluate 3 × [4 + (12 − 5) × 2]

$3 × [4 + (12 − 5) × 2]
= 3 × [4 + 7 × 2]          ← Innermost (12 − 5) = 7
= 3 × [4 + 14]              ← Inside brackets: multiplication before addition
= 3 × 18                    ← Finish inside brackets
= 54
$

Example 2: Evaluate {30 − [8 + (6 ÷ 3 × 2)]} × 4

${30 − [8 + (6 ÷ 3 × 2)]} × 4
= {30 − [8 + (2 × 2)]} × 4     ← Innermost: 6 ÷ 3 = 2
= {30 − [8 + 4]} × 4            ← Continue innermost: 2 × 2 = 4
= {30 − 12} × 4                 ← Inside square brackets: 8 + 4 = 12
= 18 × 4                        ← Inside curly braces: 30 − 12 = 18
= 72
$

⚠️ THIS IS CRITICAL — When brackets are at the same level of nesting (e.g., two separate pairs of parentheses not inside each other), evaluate them independently, in any order. But within nested brackets, ALWAYS work inside-out.

Example 3 (same-level brackets): (8 − 3) × (12 ÷ 4)

Both brackets are independent — you can evaluate either first: = 5 × 3 = 15

3. Implied Brackets — The Fraction Bar

The fraction bar (also called the vinculum) acts as an invisible bracket around the numerator and denominator.

(a + b) / (c + d) means: do a + b first, do c + d first, THEN divide.

When an expression is written vertically as a fraction, the bar groups everything above and below.

Example 1:

$  3 + 5
  -----
  2 × 4
$

This means (3 + 5) ÷ (2 × 4) = 8 ÷ 8 = 1

Without the fraction bar grouping, writing 3 + 5 ÷ 2 × 4 would give: 3 + (5 ÷ 2) × 4 = 3 + 2.5 × 4 = 3 + 10 = 13 — completely different!

Example 2:

$  12 − 2²
  -------
   4 + 1
$

= (12 − 4) ÷ (4 + 1) = 8 ÷ 5 = 1.6

When converting a horizontal expression to a fraction or vice versa, you MUST insert parentheses:

8 + 12              8 + 12
------  is NOT the same as ------  (without implied brackets)
  4                    4

8 + 12
------ = (8 + 12) ÷ 4 = 20 ÷ 4 = 5
  4

But 8 + 12 ÷ 4 = 8 + 3 = 11

Example 3: A fraction within an expression:

2 × (3 + 4) ÷ (5 − 1) + 6²

Written with a fraction bar:

$  2 × (3 + 4)
  ----------- + 6²
     (5 − 1)
$

Step-by-step:

$= 2 × 7 ÷ 4 + 36       ← Brackets and exponent
= 14 ÷ 4 + 36           ← Multiplication (left to right: 2 × 7)
= 3.5 + 36              ← Division: 14 ÷ 4
= 39.5                  ← Addition
$

4. Exponents and Roots Combined with Other Operations

When exponents and roots appear alongside multiplication, division, addition, and subtraction, follow BODMAS strictly:

1. Brackets
2. Orders (exponents and roots — evaluate these before ×, ÷, +, −)
3. Division and Multiplication (left to right)
4. Addition and Subtraction (left to right)

Example 1: 3 × 2³ + √25 ÷ 5

$3 × 2³ + √25 ÷ 5
= 3 × 8 + 5 ÷ 5        ← Orders: 2³ = 8, √25 = 5
= 24 + 1                ← Multiplication and division (left to right)
= 25                    ← Addition
$

Example 2: (5² − 3²) × 2 + √(36 + 64)

$(5² − 3²) × 2 + √(36 + 64)
= (25 − 9) × 2 + √100      ← Orders inside brackets, then addition inside square root
= 16 × 2 + 10               ← Finish brackets, evaluate square root
= 32 + 10                   ← Multiplication before addition
= 42
$

Common pitfall with −x²: Remember that −3² means −(3²) = −9, NOT (−3)² = 9. The exponent applies only to the number it's directly attached to, not to a negative sign in front.

Example 3: 10 − 2³ × (4 − 1) + √16

$10 − 2³ × (4 − 1) + √16
= 10 − 8 × 3 + 4         ← Orders and brackets
= 10 − 24 + 4             ← Multiplication: 8 × 3 = 24
= −14 + 4                 ← Subtraction: 10 − 24 = −14
= −10                     ← Addition
$

5. Roots with Complex Radicands (Expressions Under the Root)

The radical symbol √ (or ∛ for cube root) has an implied grouping over everything under it. When the radicand (expression under the root) contains operations, evaluate it first.

√(a + b) means: add a and b first, THEN take the square root.

Example 1: √(13 + 3 × 4)

$√(13 + 3 × 4)
= √(13 + 12)              ← Multiplication inside radicand before addition
= √25
= 5
$

Comparison — without parentheses: √13 + 3 × 4 would mean:

$= √13 + 12                ← The root only applies to 13
≈ 3.606 + 12
= 15.606
$

⚠️ THIS IS CRITICAL — √a + b ≠ √(a + b). The radical only covers what's directly under it. If you want the root of an entire expression, you MUST put that expression in parentheses (or under the vinculum of the radical, if it extends).

Example 2: √(9 + 16) vs √9 + √16 - √(9 + 16) = √25 = 5 - √9 + √16 = 3 + 4 = 7

These are NOT equal. The root does NOT distribute over addition.

Example 3: ∛(4² + 2 × 6)

$∛(4² + 2 × 6)
= ∛(16 + 12)               ← Exponent and multiplication inside radicand
= ∛28
≈ 3.037 (not a perfect cube, but the process is what matters)
$

6. Multi-Step Expressions — Putting It All Together

Complex expressions require systematic step-by-step evaluation. The key is to identify the highest-priority operation at each step.

Strategy for complex expressions: 1. Scan for the innermost brackets — resolve these first 2. Within each bracket (or the main expression once brackets are cleared), evaluate Orders (exponents and roots) 3. Then Division and Multiplication, left to right 4. Then Addition and Subtraction, left to right 5. Repeat at the next outer level

Example 1: Evaluate {[3² + (15 − 7) ÷ 2] × 4 − 10} ÷ 3

${[3² + (15 − 7) ÷ 2] × 4 − 10} ÷ 3
= {[3² + 8 ÷ 2] × 4 − 10} ÷ 3           ← Innermost: (15 − 7) = 8
= {[9 + 8 ÷ 2] × 4 − 10} ÷ 3              ← Order inside brackets: 3² = 9
= {[9 + 4] × 4 − 10} ÷ 3                  ← Division inside brackets: 8 ÷ 2 = 4
= {13 × 4 − 10} ÷ 3                       ← Addition inside brackets: 9 + 4 = 13
= {52 − 10} ÷ 3                           ← Multiplication: 13 × 4 = 52
= 42 ÷ 3                                   ← Subtraction: 52 − 10 = 42
= 14                                       ← Division
$

Example 2: Evaluate (√36 + 2³) ÷ (10 − 3) + 5 × (4 − 1)

$(√36 + 2³) ÷ (10 − 3) + 5 × (4 − 1)
= (6 + 8) ÷ 7 + 5 × 3                     ← Orders and all brackets
= 14 ÷ 7 + 5 × 3                           ← Finish parentheses
= 2 + 15                                   ← Division and multiplication (left to right)
= 17                                       ← Addition
$

Example 3 (with fraction bar):

$  12 + 2 × (3² − 1)
  -----------------
       √(10 + 6)
$

Step-by-step:

$Numerator: 12 + 2 × (3² − 1) = 12 + 2 × (9 − 1) = 12 + 2 × 8 = 12 + 16 = 28
Denominator: √(10 + 6) = √16 = 4
Result: 28 ÷ 4 = 7
$

7. Common Traps and How to Avoid Them

Trap 1: Left-to-right with equal-precedence operations 8 ÷ 2 × 4 Wrong (doing multiplication first because it "feels" right): 8 ÷ 8 = 1 Correct (left to right): 8 ÷ 2 = 4, then 4 × 4 = 16

Trap 2: The invisible multiplication before brackets 6 ÷ 2(1 + 2) is ambiguous notation that mathematicians avoid. Properly written: 6 ÷ 2 × (1 + 2) = 6 ÷ 2 × 3 = 3 × 3 = 9 OR, if you mean the denominator to be 2(1+2): 6 ÷ [2(1 + 2)] = 6 ÷ [2 × 3] = 6 ÷ 6 = 1 When in doubt, USE BRACKETS to make your meaning clear.

Trap 3: Forgetting that − is not the same as subtraction inside exponents −3² vs (−3)² −3² = −(3²) = −9 (exponent applies to 3 only, then negate) (−3)² = (−3) × (−3) = 9

Trap 4: Distributing incorrectly (a + b)² ≠ a² + b² (3 + 2)² = 5² = 25, but 3² + 2² = 9 + 4 = 13

Worked Examples

Example 1: Deeply Nested Expression

Evaluate: 100 − {50 − [30 − (20 − 10)]} × 2

Solution:

Work from the innermost brackets outward:

$100 − {50 − [30 − (20 − 10)]} × 2
= 100 − {50 − [30 − 10]} × 2        ← Innermost: 20 − 10 = 10
= 100 − {50 − 20} × 2               ← Inside square brackets: 30 − 10 = 20
= 100 − 30 × 2                      ← Inside curly braces: 50 − 20 = 30
= 100 − 60                          ← Multiplication before subtraction: 30 × 2 = 60
= 40                                ← Subtraction
$

Answer: 40

Example 2: Fraction Bar as Implied Grouping

Evaluate and write in simplest fraction form:

$  3² + 4 × 5
  ----------
  2³ − √25
$

Solution:

Step 1 — Numerator: 3² + 4 × 5 = 9 + 4 × 5 (exponent first) = 9 + 20 (multiplication before addition) = 29

Step 2 — Denominator: 2³ − √25 = 8 − 5 (exponent and root first) = 3

Step 3 — Result: 29 ÷ 3 = 29/3

Answer: 29/3

Example 3: Exponents, Roots, and Nested Brackets Combined

Evaluate: √[(8 ÷ 2 + 1)²] + 3 × (4² − 3²)

Solution:

$√[(8 ÷ 2 + 1)²] + 3 × (4² − 3²)
= √[(4 + 1)²] + 3 × (16 − 9)        ← Inside both brackets
= √[5²] + 3 × 7                      ← Simplify brackets
= √25 + 21                           ← Exponent and multiplication
= 5 + 21                             ← Square root
= 26                                 ← Addition
$

Answer: 26

Example 4: Word Problem Translation

Problem: "Subtract the square root of the sum of 12 and 4 from the product of 5 and the cube of 2. Then divide the result by 3."

Solution:

Step 1 — Translate to an expression: "The product of 5 and the cube of 2" = 5 × 2³ "The square root of the sum of 12 and 4" = √(12 + 4) "Subtract the second from the first" = 5 × 2³ − √(12 + 4) "Divide the result by 3" = [5 × 2³ − √(12 + 4)] ÷ 3

Step 2 — Evaluate:

$[5 × 2³ − √(12 + 4)] ÷ 3
= [5 × 8 − √16] ÷ 3       ← 2³ = 8, 12 + 4 = 16
= [40 − 4] ÷ 3             ← 5 × 8 = 40, √16 = 4
= 36 ÷ 3                   ← 40 − 4 = 36
= 12                       ← Division
$

Answer: 12

Key Terms

• Example 4: Word Problem Translation
• Roots With Complex Radicands (Expressions Under The Root)
• fraction bar
• BODMAS is universal and non-negotiable
• Correct: A) 11
• innermost
• Example 1: Deeply Nested Expression
• Correct: B) (8 + 12) ÷ √(4 × 9)
• Example 3: Exponents, Roots, And Nested Brackets Combined
• Trap 4: Distributing incorrectly
• Answer: B) 8
• Example 2: Fraction Bar As Implied Grouping

Practice Problems

(Answers are below. Try each problem before checking.)

Problem 1: Evaluate: 48 ÷ {12 − [4 + (6 − 3) × 2]}

Problem 2: Evaluate the fraction expression:

$  2 × (8 + 4)
  -----------
   3² − √16
$

Write the answer as a simplified fraction.

Problem 3: Evaluate: √(5² − 4²) × (10 ÷ 2) + 3² − 2

Problem 4: Evaluate: 2 × (3 + 5²) − √(100 − 36) ÷ 2

Problem 5: A student evaluates (4 + 6) × 2 + 3² ÷ 3 and gets 26.67. Find and correct their mistake.

Problem 6: Evaluate: {[(2³ + 1)² − 55] ÷ 13} × 5 + 10

Problem 7: Translate and evaluate: "Add the square of 6 to the cube root of 64, then divide the sum by the difference between 15 and 5."

Summary

1. BODMAS is universal and non-negotiable — it ensures everyone gets the same answer. The order is: Brackets, Orders, Division/Multiplication (L→R), Addition/Subtraction (L→R).
2. Nested brackets are resolved from the innermost pair outward. Different bracket shapes ( ), [ ], { } are for readability only — they all mean the same thing.
3. The fraction bar (vinculum) acts as an invisible bracket around both numerator and denominator — always evaluate the numerator and denominator separately before dividing.
4. The radical symbol √ only covers what's directly under it. √(a + b) ≠ √a + √b. To take the root of an expression, put the entire expression under the radical or in parentheses.
5. −x² means −(x²), not (−x)² — the exponent applies to the number, not to the negative sign. Always be explicit with brackets when there's any doubt.
6. When translating word problems into expressions, every "the sum of," "the difference between," etc., implies brackets around the corresponding operation.

Pitfalls

• Evaluating left to right ignoring precedence. 5 + 3 × 4 is 17, not 32. The multiplication happens first.
• Forgetting that division and multiplication have equal precedence. 12 ÷ 3 × 2 = 8 (left to right), not 12 ÷ 6 = 2.
• Treating a fraction bar as just a division sign without implied brackets. The expression (a+b)/(c+d) is fundamentally different from a + b / c + d.
• Applying √ to only part of the radicand. √9 + 16 ≠ 3 + 16. If the root is over the whole expression, calculate the radicand first.
• Losing track of nesting levels. When you have 3+ levels of nesting, systematic inside-out evaluation with written intermediate steps prevents errors.

Quiz

Answer each question, then read the explanation for your choice.

Q1: Evaluate: 10 + 3 × [18 − (4 + 2²)] ÷ 2

A) 28 B) 25 C) 22 D) 65

Q2: What does the fraction expression below equal?

$  5² − √9
  --------
   3 + 1
$

A) 22 B) 5.5 C) 4 D) 1

Q3: Evaluate: −4² + (10 − 3) × 2

A) −2 B) 30 C) −30 D) 2

Q4: Evaluate: 36 ÷ (2 × 3) + √(4 × 9)

A) 12 B) 60 C) 18 D) 6

Q5: Which expression correctly represents: "The sum of 8 and 12, divided by the square root of the product of 4 and 9"?

A) 8 + 12 ÷ √4 × 9 B) (8 + 12) ÷ √(4 × 9) C) 8 + 12 ÷ √(4 × 9) D) (8 + 12) ÷ √4 × 9

Q6: Evaluate: {[(5 − 2)³ ÷ 9] + 4} × 2

A) 14 B) 10 C) 18 D) 20

Q7: Evaluate: √(8 × 2 + 9) + (12 − 3²) × 2

A) 11 B) 17 C) 5 D) 23

Q8: What is the value of 2 × 3² − √(7 + 2 × 4) ÷ 3?

A) 18 B) 17 C) 6 D) 12

Next Steps

Move on to 00-11 — Laws of Indices (Exponents) for a dedicated deep-dive into the five fundamental exponent laws, negative and fractional indices, and simplifying complex exponential expressions.

Q5: Evaluate: √(3² + 4²) + (10 − 2 × 3)²

A) 21 B) 25 C) 15 D) 9

Answer: A) 21 √(9 + 16) + (10 − 6)² = √25 + 4² = 5 + 16 = 21