Brackets are small symbols that play a big role in mathematics. They group parts of expressions and equations so we know which operations to perform first. Using brackets correctly helps avoid mistakes and makes complex expressions clear and manageable.

Types of brackets
Mathematics commonly uses three kinds of brackets:

• Parentheses (round brackets): ( )
• Square brackets: [ ]
• Curly brackets: { }

These are used in the same way for grouping; they’re often nested so readers can see which group is inside another. Conventionally, we write nested brackets like { [ ( … ) ] } so each opening bracket has a matching closing bracket.

Order of operations: why brackets matter
In arithmetic and algebra there is an agreed order of operations so expressions evaluate consistently.

A common mnemonic is PEMDAS (or BODMAS in some countries):

• Parentheses / Brackets (P or B)
• Exponents / Orders (E or O)
• Multiplication and Division (MD — left to right)
• Addition and Subtraction (AS — left to right)

Brackets come first.

Any operations inside brackets must be completed before using the result in the rest of the expression. This is important because different operations produce different results depending on grouping.

Example:

• 2 × (3 + 4) = 2 × 7 = 14
• 2 × 3 + 4 = 6 + 4 = 10
  The brackets change the result by forcing the addition to happen before the multiplication.

Different operations inside brackets

Inside brackets you may find:

• Simple arithmetic (addition, subtraction, multiplication, division)
• Powers and roots
• Fractions where numerator or denominator is a grouped expression
• Algebraic expressions with variables
• Nested brackets (brackets within brackets)

When you encounter multiple operations inside brackets, you still follow the order of operations: parentheses first (innermost), then exponents, then multiplication/division, then addition/subtraction.

For nested brackets, start with the innermost pair and work outward.

What happens to a number in front of brackets?

Sometimes a number or expression appears directly in front of brackets: this typically means multiplication. For example:

• 3(2 + 5) means 3 × (2 + 5) = 3 × 7 = 21

This also applies when a variable or an expression is in front:

• (x + 2)(x − 3) means multiply the two grouped expressions.

Adding and subtracting brackets: what changes

When brackets are preceded by a plus sign, the brackets can be removed without changing the signs of the terms inside:

• +(a + b − c) = a + b − c
• 5 + (2x − 3) = 5 + 2x − 3

When brackets are preceded by a minus sign, removing them changes the sign of each term inside (you distribute the minus across the bracket):

• −(a + b − c) = −a − b + c
• 7 − (3x + 2) = 7 − 3x − 2 = 5 − 3x

This sign-change rule arises because subtracting a group is the same as adding its negative. In other words:

• a − (b + c) = a + (−1) × (b + c) = a − b − c

Distributing a number across brackets

Multiplication distributes across addition and subtraction inside brackets:

• k(a + b) = ka + kb
• k(a − b) = ka − kb

This distributes whether k is a number, variable, or more complex expression. Distribution is used to “open” or “loosen” brackets and is essential in simplifying algebraic expressions and solving equations.

The rule of “lose the brackets” (loosening brackets)
“Lose the brackets” means remove brackets by applying the appropriate rules so the expression contains no grouping symbols, or fewer of them. The core steps are:

1. If a plus sign precedes the brackets, remove the brackets without changing signs:
   • +( … ) → …
2. If a minus sign precedes the brackets, remove the brackets and change the sign of every term inside:
   • −(a + b − c) → −a − b + c
3. If a multiplication factor (number or expression) precedes the brackets, distribute it across each term inside:
   • k(a + b − c) → ka + kb − kc
4. For nested brackets, start from the innermost pair and work outward.

Examples

1. Simple removal with plus:
   4 + (2 + 3) = 4 + 2 + 3 = 9
2. Removal with minus:
   10 − (4 + 1) = 10 − 4 − 1 = 5
3. Distribute a number:
   2(3 + 4) = 2×3 + 2×4 = 6 + 8 = 14
4. Distribute a negative:
   −2(x − 5) = −2x + 10
5. Nested brackets:
   2[3(1 + 2) + 4] → evaluate innermost first:
   1 + 2 = 3 → 3(3) + 4 = 9 + 4 = 13 → 2 × 13 = 26

Tips and common mistakes

• Always look for the innermost brackets first. Solve or simplify that part before moving outward.
• Remember that subtraction of a bracket flips signs; forgetting this is a common error.
• When distributing, apply the multiplier to every term inside the bracket.
• Keep track of negative signs carefully, especially with multiple nested minuses.
• Use brackets to make expressions clearer — they prevent ambiguity.

Conclusion

Brackets are essential in mathematics because they control the order in which operations are performed. Knowing how to correctly “lose” brackets by distributing factors and changing signs keeps algebra neat and error-free. With practice, applying these rules becomes automatic and helps you simplify expressions, solve equations, and understand mathematical structure more deeply.