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Math Notes on BODMAS

BODMAS is an acronym used to remember the order of operations in mathematics. It stands for:

Brackets
Orders (i.e., powers and roots, such as squares and square roots)
Division
Multiplication
Addition
Subtraction

The BODMAS rule ensures that expressions are evaluated correctly by following these steps in the given order:

1. Brackets: Solve anything inside brackets first. This includes parentheses \(()\), square brackets \([]\), and curly brackets \(\{\}\).
2. Orders: Evaluate exponents or roots, such as \(a^2\) or \(\sqrt{b}\).
3. Division and Multiplication: Perform division and multiplication from left to right as they appear. Note that division and multiplication are of equal precedence; perform them sequentially.
4. Addition and Subtraction: Perform addition and subtraction from left to right as they appear. Like division and multiplication, addition and subtraction are of equal precedence; perform them sequentially.

Examples of Order of Operations

Example 1: Simple Calculation

Expression:
\[ 6 + 3 \times 2 \]

Solution:
1. No Brackets
2. No Orders
3. Perform Multiplication: \(3 \times 2 = 6\)
4. Addition: \(6 + 6 = 12\)

Result: \(12\)

Example 2: Expression with Brackets

Expression:
\[ (4 + 5) \times 2 \]

Solution:
1. Brackets First: \(4 + 5 = 9\)
2. No Orders
3. No Division/Multiplication in Brackets
4. Multiplication: \(9 \times 2 = 18\)

Result: \(18\)

Example 3: Expression with Orders

Expression:
\[ 3 + 2^2 \times 5 \]

Solution:
1. No Brackets
2. Orders First: \(2^2 = 4\)
3. Perform Multiplication: \(4 \times 5 = 20\)
4. Addition: \(3 + 20 = 23\)

Result: \(23\)

Example 4: Complex Expression with Multiple Operations

Expression:
\[ 8 + (6 \times 5^2 – 3) \div 3 \]

Solution:
1. Brackets First: Solve inside brackets
Orders: \(5^2 = 25\)
Multiplication: \(6 \times 25 = 150\)
Subtraction: \(150 – 3 = 147\)
So, the expression inside the brackets is \(147\)

2. Division: \(147 \div 3 = 49\)
3. Addition: \(8 + 49 = 57\)

Result: \(57\)

Example 5: Complex Expression with Nested Brackets

Expression:
\[ \frac{8 + (3 \times (2 + 4^2))}{2} \]

Solution:
1. Brackets First: Solve inside the innermost brackets
Orders: \(4^2 = 16\)
Addition Inside Brackets: \(2 + 16 = 18\)
Multiplication: \(3 \times 18 = 54\)

2. Outer Brackets: \(8 + 54 = 62\)
3. Division: \(\frac{62}{2} = 31\)

Result: \(31\)

By following the BODMAS rule, you ensure that mathematical expressions are evaluated correctly and consistently.