The Order You Must Open Brackets In
Learn to add and subtract expressions by collecting like terms, multiply and divide using the laws of indices, remove nested brackets in the right order, and substitute values.
How do you add and subtract algebraic expressions?
You add and subtract algebraic expressions by collecting like terms — grouping the terms that share the same literals to the same powers, then combining only their coefficients. Unlike terms simply stay where they are.
This page covers everything in the ICSE Class 6 Mathematics chapter on operations and substitution: the two layouts for adding, multiplying and dividing terms with indices, the fixed order for removing brackets, and finding the numerical value of an expression.
This page covers everything in the ICSE Class 6 Mathematics chapter on operations and substitution: the two layouts for adding, multiplying and dividing terms with indices, the fixed order for removing brackets, and finding the numerical value of an expression.
What are the horizontal and column methods?
Both methods give the same answer; choose whichever keeps your working tidy.
The horizontal method rewrites the sum in one line, grouping like terms:
The column method writes like terms underneath one another and adds down the columns, which is safer when there are many terms.
Subtraction needs one extra care: the minus sign applies to every term in the bracket that follows it.
Notice that became . Missing that single sign change is the most frequent error in the whole topic.
For example, if one shop has items and another has fewer, the difference is — and the sign flip is what makes the answer larger, not smaller.
The horizontal method rewrites the sum in one line, grouping like terms:
The column method writes like terms underneath one another and adds down the columns, which is safer when there are many terms.
Subtraction needs one extra care: the minus sign applies to every term in the bracket that follows it.
Notice that became . Missing that single sign change is the most frequent error in the whole topic.
For example, if one shop has items and another has fewer, the difference is — and the sign flip is what makes the answer larger, not smaller.
Formula
How do you multiply and divide algebraic terms?
Multiply the coefficients, then handle the literals using the laws of indices. For the same base, add the indices when multiplying and subtract them when dividing:
Worked example:
And dividing:
To multiply an expression by a monomial, multiply every term inside the bracket — this is the distributive property at work:
The laws of indices apply only to the same base. So cannot be simplified to a single power; it stays as .
Worked example:
And dividing:
To multiply an expression by a monomial, multiply every term inside the bracket — this is the distributive property at work:
The laws of indices apply only to the same base. So cannot be simplified to a single power; it stays as .
In what order do you remove grouping symbols?
When brackets are nested, you must work from the innermost outwards, in this fixed order:
1. bar (vinculum), the line drawn over terms
2. parentheses ( )
3. braces { }
4. square brackets [ ]
Worked example:
Remove the parentheses first:
Simplify inside the braces:
Remove the braces, changing signs because of the minus in front:
Finally multiply through: .
In real life this is just careful unpacking, like opening a parcel inside a box inside a crate — you cannot reach the innermost item first by tearing at the outside.
The sign rule is the part to watch. A minus sign before a bracket reverses the sign of every term inside it.
1. bar (vinculum), the line drawn over terms
2. parentheses ( )
3. braces { }
4. square brackets [ ]
Worked example:
Remove the parentheses first:
Simplify inside the braces:
Remove the braces, changing signs because of the minus in front:
Finally multiply through: .
In real life this is just careful unpacking, like opening a parcel inside a box inside a crate — you cannot reach the innermost item first by tearing at the outside.
The sign rule is the part to watch. A minus sign before a bracket reverses the sign of every term inside it.
How do you find the value of an expression by substitution?
Substitution means replacing each literal with a given number and then calculating. Write brackets around the substituted values so the signs and powers stay correct.
Find the value of when and :
Note that the power applies before the coefficient: you square the 2 first, then multiply by 3. Doing it the other way gives , which is wrong.
Substitution also evaluates formulas. For the perimeter of a rectangle, , with cm and cm:
Always carry the unit into the answer when the formula describes a real quantity. A bare 26 does not say whether you found a length, an area or a cost.
Find the value of when and :
Note that the power applies before the coefficient: you square the 2 first, then multiply by 3. Doing it the other way gives , which is wrong.
Substitution also evaluates formulas. For the perimeter of a rectangle, , with cm and cm:
Always carry the unit into the answer when the formula describes a real quantity. A bare 26 does not say whether you found a length, an area or a cost.
Exam tip
Exam tip: the minus sign in front of a bracket
One rule causes more lost marks in this chapter than everything else combined: a minus sign before a bracket changes the sign of every term inside, not just the first.
Students write , which is wrong. The correct expansion is .
The safe habit is to expand in two stages. First write out the bracket with every sign already flipped, then collect like terms. Never flip signs and collect terms in the same line — that is exactly where a term gets missed.
Students write , which is wrong. The correct expansion is .
The safe habit is to expand in two stages. First write out the bracket with every sign already flipped, then collect like terms. Never flip signs and collect terms in the same line — that is exactly where a term gets missed.
Did you know
Why do indices add when you multiply powers?
Because an index is only a count of how many times the base appears as a factor.
Write it out: means , which is five copies of multiplied together — so . Adding the indices is just counting the factors from both groups.
That is why the rule needs the same base: you can only pool the copies if they are copies of the same thing.
Write it out: means , which is five copies of multiplied together — so . Adding the indices is just counting the factors from both groups.
That is why the rule needs the same base: you can only pool the copies if they are copies of the same thing.
Key takeaways
Algebraic operations in 30 seconds
- Add and subtract by collecting like terms, using either the horizontal or the column layout; unlike terms remain separate.
- A minus sign before a bracket reverses the sign of every term inside it.
- Multiply coefficients and apply the laws of indices for the same base: add indices to multiply, subtract to divide.
- Multiply an expression by a monomial by distributing it over every term in the bracket.
- Remove grouping symbols innermost first — bar, then parentheses, braces and square brackets — and substitute values inside brackets, applying powers before coefficients.
You will remember all of this far better after answering five questions on it than after reading it twice.
- A minus sign before a bracket reverses the sign of every term inside it.
- Multiply coefficients and apply the laws of indices for the same base: add indices to multiply, subtract to divide.
- Multiply an expression by a monomial by distributing it over every term in the bracket.
- Remove grouping symbols innermost first — bar, then parentheses, braces and square brackets — and substitute values inside brackets, applying powers before coefficients.
You will remember all of this far better after answering five questions on it than after reading it twice.