The Simplify Calculator takes any mathematical expression, numeric or symbolic, simple or multi-step, and rewrites it in its cleanest equal form. It combines like terms, reduces fractions using the greatest common divisor (GCD), cancels common factors, simplifies radicals, applies the rules of exponents, and evaluates constants. Every result arrives with a step-by-step simplification, so you learn the method instead of copying the answer. The engine runs on symbolic algebra and supports AI-guided expression interpretation, which reads mixed notation and non-standard input that basic tools reject.
Simplify Calculator
Type any expression, whether fractions, polynomials, radicals or trig, and get its simplest form with steps, alternate forms and a clear breakdown. Instant, exact, free.
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Simplified form
Step-by-step solution
Worked examples
A library of simplifications
More than a calculator, it works as a reference. Click any example to load it instantly.
Complete guide
How to use the Simplify Calculator
The Simplify Calculator rewrites any math expression in its simplest equivalent form and shows every step. Enter a problem, pick an operation, read the worked solution. Three actions, no sign-up.
1
Enter your expression
Type the problem in the box or tap the on-screen keypad. Implicit multiplication (2x), powers (x^2), roots (sqrt(x)) and fractions (a/b) all parse naturally.
2
Pick an operation
Keep Simplify selected, or switch to Factor, Expand, Solve for, Evaluate, Derivative or Integrate. The engine applies the right rules for each task.
3
Read the result & steps
Get the simplest form, alternate forms, a decimal value and a full step-by-step solution that traces how the expression collapsed.
Entering your expression (^ for exponents, / for fractions, sqrt)
Type the expression the way you would write it on paper. Three symbols cover almost every input: ^ for exponents, / for fractions, and sqrt( ) for square roots. The on-screen keypad inserts each one for you, and the parser reads 2x as 2·x without a multiplication sign.
| To write | Type this | Reads as |
|---|---|---|
| Exponent / power | x^2 | x² |
| Fraction | 3/4 or (x+1)/(x-1) | ¾, a rational expression |
| Square root | sqrt(72) | √72 |
| Higher root | nthroot(27,3) | ∛27 |
| Implicit multiply | 2x, 3(x+1) | 2·x, 3·(x+1) |
| Trig & functions | sin(x)^2, log(100) | sin²x, log 100 |
Reading the step-by-step solution
The result panel opens with the simplest form in large type, followed by alternate forms (factored, expanded, decimal) and a numbered step list. Each step names the rule it applied, such as combine like terms, reduce by the GCD, cancel a common factor or apply an exponent rule, so you can trace the logic and find exactly where your own work diverged. Expand the step list to follow every transformation, then collapse it once the answer is clear.
What it can simplify
| Type | Example input | Simplest form |
|---|---|---|
| Numeric fractions | 48/64 | 3/4 |
| Fraction arithmetic | 2/3 - 3/2 + 1/4 | −7/12 |
| Rational expressions | (x^2-4)/(x+2) | x − 2 |
| Polynomials | 2x + 3x - x | 4x |
| Radicals | sqrt(72) | 6√2 |
| Exponents | 4 + (2+1)^2 | 13 |
| Trigonometry | sin(x)^2 + cos(x)^2 | 1 |
| Factorials | 5! | 120 |
Worked examples
Worked examples by type
Five expression types cover most homework: algebraic expressions, fractions, radicals, exponents and rational expressions. Each one follows a fixed method.
Simplify algebraic expressions (combine like terms)
Add the coefficients of terms that share the same variable and power. Only like terms combine.
4x + 7x − 2x→9x- Group the like terms: all three carry
x. - Add the coefficients: 4 + 7 − 2 = 9.
- Result: 9x. When higher terms appear, such as a difference of squares, perfect square trinomials or quadratic factorization, the engine factors first, then reduces.
Simplify fractions (reduce using GCD)
Find the greatest common divisor (GCD) of the numerator and denominator, then divide both by it to reach lowest terms.
18/24→3/4- List the factors: GCD of 18 and 24 is 6.
- Divide top and bottom by 6: 18 ÷ 6 = 3, 24 ÷ 6 = 4.
- Result: 3/4, with nothing left to cancel.
Simplify radicals (square roots)
Split the number under the square root into its largest perfect-square factor, then pull that factor out.
sqrt(72)→6√2- Factor 72 as 36 × 2, where 36 is a perfect square.
- Rewrite: √72 = √36 · √2.
- Take the root of 36: 6√2. Exact radical form keeps full precision for geometry and physics formulas.
Simplify expressions with exponents
Apply the rules of exponents: add powers when multiplying the same base, subtract when dividing, multiply when raising a power to a power.
x^5 · x^2 / x^3→x⁴- Multiply same base: x⁵ · x² = x⁷ (5 + 2).
- Divide same base: x⁷ / x³ = x⁴ (7 − 3).
- Result: x⁴. Negative and zero exponents fold in by the same laws.
Simplify rational expressions (with restrictions)
Factor the numerator and denominator, cancel the shared factor, and record the value that would make the original denominator zero.
(x²−4)/(x+2)→x − 2, x ≠ −2- Factor the numerator as a difference of squares: x² − 4 = (x − 2)(x + 2).
- Cancel the common factor (x + 2) from top and bottom.
- State the restriction: the original denominator was zero at x = −2, so x ≠ −2 stays attached to the answer. The restriction marks where the simplified form and the original are not equal.
Definition
What does "simplify" mean in math?
To simplify means to rewrite an expression in its shortest, clearest form without changing its value. 2x + 3x becomes 5x; 6/12 becomes 1/2; √50 becomes 5√2. Same value, fewer parts.
Simplify vs solve
Simplifying rewrites an expression; solving finds a value. An expression has no equals sign, so it has nothing to solve. 2x + 4x simplifies to 6x and stops there. An equation has an equals sign, and the equation solver isolates the variable: x² − 4 = 0 solves to x = 2 or x = −2. Many problems simplify first, then solve, because fewer terms make the variable easier to isolate.
Simplify vs expand vs factor
Simplify
Reduce to the cleanest equal form.
2(x+3) + 4x → 6x + 6Expand
Multiply grouped terms out.
(2x+3)(x−5) → 2x² − 7x − 15Factor
Rewrite as a product of factors.
x² − 5x + 6 → (x−2)(x−3)Expanding and factoring are opposite moves, and simplifying may use either one along the way. The Simplify Calculator exposes all three as separate operations so you control which form you want.
What counts as "simplest form"?
An expression is in simplest form when four conditions hold: like terms are combined, fractions are reduced to lowest terms by their GCD, radicals carry no remaining perfect-square factor, and no parentheses can be dropped without changing the value. A rational expression is fully simplified only when the numerator and denominator share no common factor and the restriction is recorded.
The method
Simplification rules & order of operations
Four rule sets drive every simplification: order of operations, combining like terms, the rules of exponents, and the fraction & radical rules.
PEMDAS (order of operations)
Complex expressions evaluate in a fixed order of operations, written as PEMDAS in the US and BODMAS elsewhere. The two names describe the same sequence.
- PParentheses: resolve grouped terms first.
- EExponents: apply powers and roots next.
- MDMultiplication & division: left to right.
- ASAddition & subtraction: left to right, last.
Order changes the answer: 3 + 4 × 2 is 11, not 14, because multiplication runs before addition. 4 + (2+1)^2 is 13, since parentheses give 3, the exponent gives 9, then 4 + 9.
Combining like terms
Combine like terms
Terms with the same variable and power add through their coefficients: 2x + 3x = 5x. Unlike terms such as 3x and 2x² stay separate.
Cancel common factors
Divide a fraction's numerator and denominator by their GCD until nothing shared remains. Cancel factors, never added terms.
Exponent rules
| Rule | Form | Example |
|---|---|---|
| Product rule | aᵐ · aⁿ = aᵐ⁺ⁿ | x² · x³ = x⁵ |
| Quotient rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | x⁵ ÷ x² = x³ |
| Power rule | (aᵐ)ⁿ = aᵐⁿ | (x²)³ = x⁶ |
| Zero rule | a⁰ = 1 | 7⁰ = 1 |
| Negative rule | a⁻ⁿ = 1 / aⁿ | x⁻² = 1/x² |
Fraction & radical rules
Reduce fractions
Divide top and bottom by the GCD: 18/24 = 3/4. For algebraic fractions, factor first, then cancel.
Reduce radicals
Pull out the largest perfect-square factor: √72 = √(36·2) = 6√2. Rationalize denominators when a root sits underneath.
Watch out
Common mistakes when simplifying
Five errors account for most wrong answers. A rule-based calculator avoids each one by applying validated algebraic logic.
Incorrect factor cancellation
Cancelling terms that are added, not multiplied, breaks the value.
Wrong (x² + x)/x = x² + 1
Right x(x + 1)/x = x + 1
Sign errors
A minus in front of parentheses applies to every term inside.
Wrong 5 − (x + 3) = 5 − x + 3
Right 5 − (x + 3) = 5 − x − 3 = 2 − x
Partial simplification
Stopping early leaves the expression more complex than it needs to be.
Wrong 18/24 = 9/12
Right 18/24 = 3/4
Exponent misrules
Multiplying bases adds exponents; it does not multiply them.
Wrong x² · x³ = x⁶
Right x² · x³ = x⁵
Ignoring order of operations
PEMDAS is not optional, so multiplication runs before addition.
Wrong 3 + 4 × 2 = 14
Right 3 + 4 × 2 = 11
Combining unlike terms
Different variables or powers cannot merge into one term.
Wrong 3x + 2x² = 5x³
Right 3x + 2x² stays as is
The payoff
Why simplifying expressions matters
Simplification is not a formality. Fewer terms mean fewer chances to slip, clearer patterns, and faster work in every field that uses formulas.
In algebra & equation solving
Simplified expressions make isolating a variable easier. A reduced equation has fewer terms to track, so each step toward the solution carries less risk of a sign or factor error. Simplification also exposes structure, such as a common factor, a difference of squares or a perfect square trinomial, that points to the next move. Checking work gets faster too: if your form does not match the calculator's simplest form, that mismatch flags a mistake before it spreads.
In engineering, finance & real life
Reduced formulas read clearly and cut substitution errors across applied fields:
- Engineering formulas: fewer terms lower the chance of a wrong substitution under load or stress calculations.
- Finance calculations: interest formulas and ratios usually simplify before evaluation, during tax filing season or rate comparisons.
- Physics problems: distance, velocity and force equations rely on algebraic reduction to stay readable.
- Data modeling: statistical and exponential expressions need simplification before they can be interpreted.
- Everyday math: scaling a recipe, splitting a bill or comparing two plans all reduce to simpler arithmetic.
Who it's for
Who uses a Simplify Calculator
Three groups reach for it most: students working through coursework, teachers building and checking material, and parents helping at home.
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Students (pre-algebra to calculus)
From combining like terms in pre-algebra to simplifying rational expressions before a derivative limit, students use the step-by-step solution to check homework, practice new methods, and find exactly where an answer went off course. The blog adds more worked examples and study guides.
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Teachers & tutors
Teachers and tutors generate clean worked examples, confirm answer keys, and show the reasoning behind each step on screen. Rule-based output keeps every example consistent across a lesson or worksheet.
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Parents helping with homework
Parents who last touched algebra years ago follow the labeled steps to support a child without re-teaching themselves the whole subject. The breakdown explains the why, not just the final answer.
Questions
Frequently asked
Is the simplify calculator free?
Yes, the simplify calculator is completely free with no sign-up. It runs entirely in your browser, so your work stays private on your device and there is no usage limit.
Does it show step-by-step solutions?
Yes. Every result includes a numbered step-by-step simplification plus alternate forms (factored, expanded and decimal), so you see the full method, not only the final answer.
Can it simplify fractions, radicals, and exponents?
Yes. The calculator reduces fractions using the GCD, simplifies radicals by extracting perfect squares, and applies the rules of exponents. It also handles algebraic expressions, rational expressions and trigonometric identities.
What's the difference between simplify and solve?
Simplify rewrites an expression in its cleanest equal form, such as 2x + 4x → 6x. Solve finds the value of a variable in an equation, such as x² − 4 = 0 → x = ±2. Simplify has no equals sign to resolve; solve does.
Why does it show restrictions like x ≠ 1?
A restriction marks a value that would make the original denominator zero. When a rational expression cancels a factor, the simplified form equals the original everywhere except that point, so the restriction stays attached to keep the two expressions mathematically equal.
How do I type exponents and square roots?
Use ^ for exponents (x^2), sqrt( ) for square roots (sqrt(72)), and / for fractions. The on-screen keypad inserts each symbol, and 2x is read as 2·x.
Does it work on mobile?
Yes. The calculator runs in real time on phones, tablets and computers with no installation. The on-screen keypad covers every symbol you need on a touch screen.
Ready when you are
Try the Simplify Calculator now
Enter any expression, whether fractions, polynomials, radicals, exponents or trig, and get its simplest form with a full step-by-step solution. Instant, exact, free.
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