Lesson 1.6

Order of Operations (Advanced)

Now we level up: nested grouping symbols, fraction bars as hidden parentheses, and absolute value bars. When groups live inside groups, work from the innermost pair outward.

Introduction

In Lesson 1.5 we handled one layer of parentheses. Real-world formulas often have brackets inside brackets and fraction bars that act as invisible parentheses.

Past Knowledge

You can apply PEMDAS to expressions with one level of parentheses.

Today's Goal

Simplify expressions with nested grouping symbols, fraction bars, and absolute value.

Future Success

Advanced equations and formulas regularly use these nested structures.

Key Concepts

1. Types of Grouping Symbols

SymbolNameExample
Parentheses
Brackets
Fraction barDivision line
Absolute value

2. The Inside-Out Rule

Always simplify the innermost grouping symbol first, then work outward.

3. Fraction Bars Are Parentheses

A fraction bar groups everything on top and everything on bottom separately. Simplify the numerator and denominator independently, then divide.

Worked Examples

Example 1: Nested Parentheses

Basic

Simplify .

1

Innermost First

→

2

Next Group

→

3

Final Multiply

Example 2: Fraction Bar

Intermediate

Simplify .

1

Numerator

2

Denominator

3

Divide

Example 3: Absolute Value + Nested

Advanced

Simplify .

1

Absolute Value

2

Inner Parentheses

→

3

Brackets

→

4

Finish

Common Pitfalls

Ignoring the Fraction Bar

In , you must add first. Don't divide and then add .

Starting from the Outside

With nested groups like , always start with the innermost pair, not the outer brackets.

Real-Life Applications

Every programming language (Python, JavaScript, C++) follows PEMDAS with nested grouping. When a software engineer writes a complex calculation, getting the nesting wrong can crash an entire application or produce dangerously wrong results — like miscalculating a rocket trajectory.

Practice Quiz

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