Revise how to read, tidy and stretch algebraic expressions — collecting like terms, expanding brackets and using the index laws — so algebra feels neat and predictable by the time you sit your Checkpoint.
At a glance
An expression is letters and numbers joined by operations — no equals sign.
A term is one chunk separated by + or −, like 5x, −3y or 7.
Like terms have the same letters with the same powers — only their coefficients change.
Collect like terms by adding or subtracting their coefficients: 5x+2x=7x.
Expand a bracket by multiplying every inside term by the outside factor.
Index laws: xa×xb=xa+b and .
Anything to the power 0 is 1, and (xa)b=x.
Tidy in one direction — expand first, then collect — and check the sign of every term.
What you’ll learn
Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.
Use letters to write and read algebraic expressions confidently.
Simplify expressions by collecting like terms, including with negatives.
Expand single brackets and multiply terms with the same base.
Apply the index laws to multiply, divide and raise powers in algebra.
The language of algebra
An expression is made of terms — each term has a coefficient and a letter part.
An algebraic expression is a mix of numbers, letters and operations — for example 5x+3y−7. There is no equals sign, so it is not something you solve; it is something you read, tidy and sometimes stretch out.
The pieces between the + or − signs are called terms. So has three terms: , and . Each letter term is built from a (the number in front) and a (the letter). In the coefficient is and the variable is .
Collecting like terms
Like terms share the same letters with the same powers — only their coefficients change.
Like terms are terms that have exactly the same letter part. So 5x and 2x are like terms, and so are 3xy and −4xy. But and are like terms — the powers are different — and neither are and .
Expanding a single bracket
Multiply every term inside the bracket by the factor outside.
To expand (or multiply out) a bracket, multiply every term inside by the factor sitting outside. This works because of the distributive law: a(b+c)=ab+ac.
Index laws in algebra
The same index rules from arithmetic carry straight into algebra.
An index (or power) tells you how many times a base is multiplied by itself, so x4=x×x×x×x. Three rules cover almost everything you need.
Multiplying with the same base — add the indices: . So .
Putting it all together
Most tidy-up questions mix all four skills — work in the same order every time.
A typical "simplify" question pulls in several tools at once. A reliable order is:
Expand any brackets first.
Apply the index laws to combine same-base powers.
Collect like terms to finish.
Take 4(2x+3)−2(x−5). Expand both brackets carefully: . There are no powers to combine, so collect: .
Where you'll use this next
Tidy algebra is the launchpad for almost every later topic.
Fluent expression-work unlocks a lot of the maths that comes next:
Solving equations starts with simplifying both sides — collect like terms, expand brackets, then balance.
Formulae use exactly the same rules, just with more letters: substituting and rearranging both need confident algebra.
Quadratic expressions build directly on expanding and the index laws (x×x=x2).
Graphs and sequences describe patterns with expressions, so tidy notation pays off there too.
If a later topic feels slow, it is often a like-terms slip or a missed sign that needs a quick refresh. Come back to this guide whenever you need a steady reset — accuracy beats speed every time.
Equations start with simplifying both sides.
Quick recap
An expression is letters and numbers joined by operations — no equals sign.
Like terms share the same letter parts; combine only their coefficients.
Carry every sign with its term as you regroup or expand.
Expand a bracket by multiplying each inside term by the outside factor.
Add indices when multiplying, subtract when dividing, multiply for a power of a power.
Anything to the power 0 is 1; an invisible index is 1.
Expand first, then collect — and write one step per line.
Read every expression as a list of terms, signs and all.
A few quiet conventions worth knowing: 1x is always written as x, −1x as −x, and x×y as xy. A term with no letter — like the −7 above — is called a constant. Read every expression slowly the first time so you do not lose track of a sign.
An expression has no equals sign — it is read, not solved.
A term is a single chunk separated by + or −.
Each letter term has a coefficient and a variable.
1x is written as x; x×y is written as xy.
5x
5x2
not
5x
5y
To simplify, you add or subtract the coefficients of like terms and leave the letter part untouched.
5x+2x=7x
9y−4y=5y
7a+3b−2a+5b=5a+8b
Group like terms first — a slip on the sign is the easiest mistake to avoid.
Two tips speed this up. First, carry each sign with its term when you regroup — 7a−2a, not 7a+2a. Second, write the simplified terms in a sensible order: letters in alphabetical order, then the constant last, so 3a+4b−2 reads cleanly.
Like terms share the same variables raised to the same powers.
Add or subtract only the coefficients — leave the letters alone.
Carry the sign with each term as you regroup.
Write the tidy answer alphabetically with the constant last.
3
(
x
+
4)=
3x+
12
−2(y−5)=−2y+10
x(x+7)=x2+7x
Every inside term gets multiplied — none is left behind.
After expanding, collect like terms if any new ones appear. For example 4(x+3)+2(x−5)=4x+12+2x−10=6x+2. Expand first, then tidy — one job at a time, and the signs stay under control.
Multiply each term inside the bracket by the factor outside.
A negative factor flips the sign of every inside term.
Expand first, then collect like terms separately.
x(x+7)=x2+7x — keep the index when multiplying by x.
xa×xb=xa+b
x5×x2=x7
Dividing with the same base — subtract the indices: xbxa=xa−b. So x3x7=x4.
Power of a power — multiply the indices: (xa)b=xab. So (x3)2=x6.
Same base is the rule — different bases stay separate.
Two extra facts close the gaps. Anything (except zero) to the power 0 is 1, so x0=1. And a number with no index shown has an invisible index of 1, so x=x1. When a term has both a coefficient and a variable — 3x2×4x5 — multiply the coefficients separately from the indices: 3×4=12, x2+5=x7, giving 12x7.
Add indices when multiplying same-base powers.
Subtract indices when dividing same-base powers.
Multiply indices for a power of a power.
Anything to the power 0 is 1; an invisible index is 1.
8x+12−2x+10
(8x−2x)+(12+10)=6x+22
Try one with powers: 2x×3x2+5x3−x×x2. Use the index law on each product first: 2x×3x2=6x3 and x×x2=x3. The expression becomes 6x3+5x3−x3=10x.
Two final habits keep you accurate. Write one step per line — squashing two steps together is where sign errors hide. And sense-check the answer — if the original had only x terms and constants, the simplified version should too.
Always expand brackets before collecting.
Apply index laws to same-base products and quotients.
Collect like terms once everything else is tidied.
One step per line — sign slips love crowded working.
Formulae use the same rules with more letters.
Quadratics build on expanding and indices.
Sequences and graphs use expressions to describe patterns.
+
3y)
Step 2
Combine the coefficients of each group.
6x−2x=4x,4y+3y=7y
5×2x=10x,5×(−3)=−15
Step 2
Write the two products as one expression.
10x−15
Answer
10x−15
5
Step-by-step solution
Step 1
Multiply the coefficients separately from the indices.
4×3=12
Step 2
Add the indices since the bases are both x.
x3×x5=x3+5=x8
Step 3
Bring the two parts together.
4x3×3x5=12x8
Answer
12x8
−
2)
Step-by-step solution
Step 1
Expand the first bracket carefully.
3(2x+5)=6x+15
Step 2
Expand the second bracket — the −4 multiplies both inside terms.
−4(x−2)=−4x+8
Step 3
Write everything in one line.
6x+15−4x+8
Step 4
Collect like terms.
(6x−4x)+(15+8)=2x+23
Answer
2x+23
2
+
5(x3−
2x)−
x×
x2
Step-by-step solution
Step 1
Apply the index law to each product separately.
2x×3x2=6x3,x×x2=x3
Step 2
Expand the bracket.
5(x3−2x)=5x3−10x
Step 3
Write the whole expression out.
6x3+5x3−10x−x
Step 4
Collect the x3 terms; the −10x stands alone.
(6+5
Answer
10x3−10x
3y
−7
2
−2(y−5)=−2y+10
How to avoid it
Multiply the outside factor by every inside term, signs and all. −2×−5=+10.
▼
Why it happens
The three index rules get mixed up under pressure.
How to avoid it
Multiply same-base powers by adding the indices: x3×x2=x3+2=x5.
3x−3
▼
Why it happens
The minus only seems to apply to the first inside term.
How to avoid it
A minus sign in front of a bracket flips every inside sign: 4x−(x−3)=4x−x+3=3x+3.
▼
Why it happens
The invisible coefficient in x is overlooked.
How to avoid it
Read x as 1x when collecting: 1x+3x=4x. Same idea for −x, which is −1x.