To simplify an expression, combine like terms, remove parentheses, and apply order of operations until no step remains.
When you reduce an algebraic expression, you make the same quantity easier to read and faster to use. The goal is clarity without changing value. This guide gives a clear path, common moves, and traps to avoid, with worked problems that build skill.
Core Method: Step-By-Step Simplifying
Use this steady routine any time you face a messy line of symbols. It stays the same for integers, fractions, and variables.
- Clean formatting: rewrite the line with tidy spacing and clear fractions.
- Strip grouping: remove parentheses and brackets by distributing or by collapsing inside-out.
- Apply order of operations: handle powers, then multiplication and division from left to right, then addition and subtraction.
- Combine like terms: add or subtract terms that share the same variable parts and exponents.
- Reduce fractions: factor numerators and denominators and cancel common factors.
- Scan for finish: no like terms left, no parentheses for distribution, no extra common factors.
Common Moves You’ll Use Often
The table below acts like a quick deck of moves you’ll reach for again and again.
| Move | Meaning | Quick Example |
|---|---|---|
| Like terms | Match exact variable part | 3x + 5x → 8x |
| Distribution | Multiply across parentheses | 2(a + 3) → 2a + 6 |
| Factoring | Pull out a common factor | 6y + 9 → 3(2y + 3) |
| Cancel factors | Reduce a fraction | (4x)/(2) → 2x |
| Power rules | Add exponents when multiplying like bases | x² · x³ → x⁵ |
| Zero/one | Use 0 and 1 smartly | x + 0 = x; 1·x = x |
| Negative signs | Attach sign to factor or term | −(a − b) → −a + b |
Simplifying An Algebraic Expression Step By Step
Run the method on these cases. Each one keeps the value but trims the clutter.
Case 1: With Parentheses And Like Terms
Problem: 2(x + 4) − 3x + 5
Work: distribute → 2x + 8 − 3x + 5 → combine like terms → (2x − 3x) + (8 + 5) → −x + 13
Result: −x + 13
Case 2: Fractions And Factors
Problem: (6x² − 9x) / (3x)
Work: factor top → 3x(2x − 3) / (3x) → cancel 3x → 2x − 3
Result: 2x − 3
Case 3: Exponents With Same Base
Problem: x³ · x · x²
Work: add exponents → x^(3 + 1 + 2) → x⁶
Result: x⁶
Case 4: Distribution Over A Difference
Problem: −2(3y − 5) + 4
Work: distribute the −2 → −6y + 10 + 4 → combine numbers → −6y + 14
Result: −6y + 14
Case 5: Mixed Operations With Order
Problem: 5 − 2² · 3 + 4
Work: powers first → 5 − 4 · 3 + 4 → multiply → 5 − 12 + 4 → add/subtract left to right → −3
Result: −3
Order Of Operations, In Plain Language
The common rule set for order goes by a short mnemonic. Move through grouping, powers, multiplication and division from left to right, then addition and subtraction from left to right. This keeps everyone on the same page.
You can read a friendly reference for these steps on order of operations. For skill practice and videos, see the Khan Academy review.
How To Spot Like Terms Fast
Two terms are “like” when they share the same variable part and the same exponents. Coefficients can differ. Constants are like with constants.
- 3x and −7x are like → combine to −4x.
- 2x² and 5x are not like → leave them separate.
- 4 and −11 are like → combine to −7.
When a term hides inside parentheses, clear the grouping first, then combine.
Distributing Without Slips
Multiplying across parentheses sends the factor to every term inside. The sign goes with the factor. Missing a term is a common slip, so run a quick mental checklist: “hit each term once.”
Example: −3(2a − a² + 5) → −6a + 3a² − 15
Fractions: From Busy To Clear
Many expressions drop into place once you factor parts of a fraction and cancel common factors. Only cancel factors that multiply the entire top and bottom.
Good: (5x)/(10) → (5·x)/(5·2) → x/2
Not allowed: (x + 5)/x → you cannot cross out x since it’s not a factor of the entire top.
Exponents: The Few Rules You Need
With the same base, multiplication adds exponents and division subtracts them. A power raised to a power multiplies exponents. A negative exponent flips to a reciprocal. These four ideas carry most tasks.
- x^m · x^n = x^(m + n)
- x^m / x^n = x^(m − n)
- (x^m)^n = x^(mn)
- x^−n = 1/x^n
Numbers First, Variables Second
When both numbers and variables appear, trim the numbers first. Then handle the variable parts. This split view keeps steps short and neat.
Demo: 12x²y / 18xy² → reduce numbers → (12/18)·(x²/x)·(y/y²) → (2/3)·x·(1/y) → 2x/(3y)
Another demo: (15a³b² − 10a²b) / 5ab → factor top → 5ab(3a²b − 2a) / 5ab → cancel 5ab → 3a²b − 2a
Worked Set: From Messy To Neat
Problem A
Given: 4(2x − 3) − (x − 5) + 7
Steps: 8x − 12 − x + 5 + 7 → combine like terms → (8x − x) + (−12 + 5 + 7) → 7x + 0 → 7x
Problem B
Given: (9y² − 6y) / (3y)
Steps: factor top → 3y(3y − 2) / (3y) → cancel 3y → 3y − 2
Problem C
Given: (a²b)(ab³) / (a³b)
Steps: combine tops → a³b⁴ / (a³b) → subtract exponents → b³
Problem D
Given: 2/(x) + 3/(2x)
Steps: common denominator 2x → (4 + 3)/(2x) → 7/(2x)
Signs And Subtraction
Subtraction adds a negative. That tiny idea clears many headaches. Switch “minus a group” into “add the opposite,” then distribute.
Quick demo: 5 − (2x − 7) → 5 + (−2x + 7) → −2x + 12
Longer Walk-Throughs
Extended Problem 1
Given: (3x − 2)(x + 5) − (x − 4)(x + 1)
Steps: expand both → (3x² + 15x − 2x − 10) − (x² + x − 4x − 4) → 3x² + 13x − 10 − x² − x + 4x + 4 → combine like terms → (3x² − x²) + (13x − x + 4x) + (−10 + 4) → 2x² + 16x − 6
Extended Problem 2
Given: (8 − 2x)/(4) + (x − 6)/(2)
Steps: split first fraction → 8/4 − (2x)/4 + (x − 6)/2 → 2 − x/2 + x/2 − 3 → cancel x/2 terms → −1
When To Stop
Learners often ask when a line counts as “done.” Stop when these checks pass: no like terms remain, every fraction is reduced, exponents follow the rules above, and no open parentheses need a distribution pass. If two forms are equal and both pass the checks, either works.
Practice Drills You Can Try Now
Test yourself with this short set. Answers sit below the table so you can self-check without scrolling far.
| Drill | Target | Hint |
|---|---|---|
| 3(2x − 5) + 4x | Combine like terms | Distribute first |
| (8y³ · y) / (2y²) | Exponent rules | Subtract exponents |
| (12 − 3x) / 6 | Fraction reduce | Split the fraction |
| −(a − 4) + 2(a + 1) | Signs & distribution | Add the opposite |
| (x² − 9)/(x − 3) | Factoring | Difference of squares |
Drill Answers
- 3(2x − 5) + 4x → 6x − 15 + 4x → 10x − 15
- (8y³ · y) / (2y²) → 8y⁴ / 2y² → 4y²
- (12 − 3x) / 6 → 2 − (x/2)
- −(a − 4) + 2(a + 1) → −a + 4 + 2a + 2 → a + 6
- (x² − 9)/(x − 3) → (x − 3)(x + 3)/(x − 3) → x + 3
Pitfalls And Fixes
Most slips fall into a short list. Spot them and your work speeds up.
| Mistake | Why It Fails | Fix |
|---|---|---|
| Cancel across sums | Top isn’t a single factor | Factor first, then cancel |
| Forget a term in distribution | Factor didn’t hit every term | Tick off each term |
| Mix like/unlike terms | Variable parts don’t match | Match exponents and letters |
| Skip order rules | Wrong sequence of steps | Run the left-to-right pass |
| Sign drops | Minus turns to plus mid-line | Bundle sign with factor |
Speed Tips That Still Keep Accuracy
- Rewrite messy work before you start; a tidy line cuts errors.
- Circle like terms in matching colors; then combine in one sweep.
- Split complex fractions into pieces, reduce, then merge again.
- On a test, leave space between steps; mistakes pop out faster.
- Do a last scan: grouping, powers, multiply/divide, add/subtract.
When To Leave Factored Or Expanded
Both forms are fine in many tasks. If the next step asks you to plug in a value, expanded form reads faster. If the next step asks for solving or reducing a fraction, a factored line shines.
Mini Reference: Patterns Worth Memorizing
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- a² − b² = (a − b)(a + b)
These patterns speed up checks and factoring passes.
Checklist Before You Submit An Answer
- All parentheses cleared or factored on purpose?
- Order of operations applied in the right sequence?
- Like terms combined?
- Fractions reduced?
- Signs carried through each step?
Where To Practice More
Short, daily work beats long cramming. Set a five-minute drill block, pick ten lines, and time your run. Track errors and target the same type the next day. Free sets and videos live on the two links above.