Simplify Calculator – Simplify Fractions, Radicals and Ratios Step by Step

One of each, fully worked

The three jobs this calculator does, on the three examples students meet first. Every one of them is a live default above — type the numbers in and the steps rebuild themselves.

Keep going with the neighbours

Simplifying is one move in a bigger game — these tools pick up where it leaves off.

How it works

Three moves cover every mode on this page — find what is shared, take it out, prove nothing is left.

The math behind the default example

Two questions worth slowing down for

When can you stop — and why does dividing by the GCD always land in one jump?

Prime factors: the other way to the same GCD

Euclid’s method is the fastest way to a GCD, but prime factorization is the way that shows why the GCD is what it is. Every whole number breaks into primes in exactly one way, and the GCD is simply the primes two numbers share, taken at the lowest power that appears in both. The prime-factor box under the step-by-step working on this page does this breakdown for your current numbers; this section explains how to read it and how to do it by hand.

Start with 24 and 36. Divide by the smallest prime that fits, and keep going until only primes remain. For 24: 24 ÷ 2 = 12, 12 ÷ 2 = 6, 6 ÷ 2 = 3, and 3 is prime — so 24 = 2^3 × 3. For 36: 36 ÷ 2 = 18, 18 ÷ 2 = 9, 9 ÷ 3 = 3 — so 36 = 2^2 × 3^2. The shared primes are 2 and 3. The lowest power of 2 in either number is 2^2, and the lowest power of 3 is 3^1, so the GCD is 2^2 × 3 = 12. That is exactly the number Euclid’s chain gives, reached by a completely different road — and when two methods agree, you can trust the answer.

The same idea works when the numbers are less friendly. Take 150 and 100, the improper-fraction example from the default walkthrough. 150 = 2 × 3 × 5^2 and 100 = 2^2 × 5^2. They share 2 (lowest power 2^1) and 5 (lowest power 5^2), but 3 appears only in 150, so it contributes nothing. The GCD is therefore 2 × 25 = 50, and 150/100 reduces to 3/2. Notice what the factorization tells you that Euclid’s chain does not: the leftover factors, 3 and 1, are exactly the simplified numerator and denominator, because dividing by the GCD strips out precisely the shared part and leaves the unshared part behind.

Prime factors also explain radicals. A square root can only release a prime in pairs: two copies of a prime under a square root make one copy outside it, and any prime left single stays inside. For √72, the factorization is 72 = 2^3 × 3^2. The 3^2 gives one 3 outside and nothing left over; the 2^3 gives one pair (one 2 outside) and one 2 left over. Outside: 2 × 3 = 6. Inside: 2. So √72 = 6√2, with the factorization showing every move. For a cube root the grouping is in threes instead: 108 = 2^2 × 3^3, so the 3^3 gives one 3 outside, the 2^2 cannot make a triple and stays inside as 4, and ∛108 = 3∛4.

When should you use factorization instead of Euclid’s method? For small numbers, factorization is often the quicker mental route, and it doubles as a check on the Euclid chain. For large numbers it becomes slow — finding prime factors of a big number is genuinely hard work, while Euclid’s repeated remainders stay fast — which is why the calculator leads with Euclid and offers factorization as the working underneath. Either way, the test at the end is the same: the simplified parts must share no factor except 1.

Mixed numbers and decimals: the same answer in three clothes

A simplified fraction can be worn three ways — as an improper fraction, as a mixed number, and as a decimal — and none of them changes the value. The calculator shows all three for fraction mode: the simplest form in the headline, the mixed number in the breakdown table, and the decimal and percent as tiles. Knowing when to use each form matters more than knowing how to convert between them, because each one is the right answer in a different setting.

Converting an improper fraction to a mixed number is one division. Take 150/100, which simplifies to 3/2. Divide 3 by 2: it goes once, with remainder 1. The quotient is the whole-number part and the remainder goes over the same denominator, giving 1 1/2. The same move on 7/3 gives 2 1/3, since 7 ÷ 3 is 2 remainder 1. If there is no remainder, the fraction was a whole number in disguise: 36/12 simplifies to 3/1, which is just 3, and the calculator says so rather than dressing it up as a mixed number. A proper fraction never becomes a mixed number at all — 2/3 is already smaller than 1, so there is no whole part to extract, and the breakdown table says so plainly.

Negative fractions follow the same rule with the sign carried once, on the whole number. −7/3 simplifies to −2 1/3, meaning negative two and one third — not −2 plus a positive third, which would be a different number. The calculator puts the sign on the numerator throughout, so the mixed number it reports always means the negative of the positive mixed number, which is the convention textbooks use.

The decimal form is where exactness ends and rounding begins, and the page is honest about the boundary. The fraction itself is exact — 2/3 is exactly two thirds — but its decimal, 0.666667, is rounded to 6 places, and its percent, 66.6667%, to 4 places. For 7/3 the decimal is 2.333333 and the percent 233.3333%, both rounded, both slightly short of the true value. That is why a simplified fraction is the answer to keep for further working: decimals rounded early and then multiplied again drift further from the truth with every step, while the fraction carries no error at all. Use the decimal when a measurement, a price or a comparison needs it; keep the fraction when the next step is more mathematics.

A useful habit is to check the three forms against each other. For 150/100: the simplest form 3/2, the mixed number 1 1/2 and the decimal 1.5 must all agree, and they do — 1 1/2 is one and a half, and 3 ÷ 2 is 1.5. If a conversion ever disagrees with the other two, the conversion is the suspect, not the fraction. That cross-check catches the most common slip, which is writing the remainder over the wrong denominator — 7/3 is 2 1/3, never 2 1/7.

Radicals past the square root: cubes, coefficients and finished forms

Square roots get most of the attention, but the same pull-out rule governs cube roots and radicals with coefficients — only the size of the group changes. For a square root you hunt for pairs; for a cube root you hunt for triples; and a coefficient in front multiplies whatever comes out. Once those three ideas are in place, every radical on this page follows the same script.

Start with a coefficient, because it is the part people most often mishandle. In 2√72, the 2 is not under the root, so it waits outside while the root is simplified: √72 becomes 6√2, and then 2 × 6 = 12, giving 12√2. The coefficient never enters the root and the root never swallows the coefficient. If the root simplifies completely — a perfect square or cube — the coefficient multiplies a plain number instead: 2√36 = 2 × 6 = 12. And if the coefficient is 0, the whole expression is 0 before any root work begins, which is why the calculator reports 0 immediately for that case.

Cube roots reward the same patience. ∛16 looks stuck until you factor: 16 = 2^4 = 8 × 2, and 8 is a perfect cube, so ∛16 = 2∛2. ∛108 = 3∛4, because 108 = 27 × 4 and 27 is 3^3. The finished-form test is identical in spirit to the square case: a cube root is fully simplified when the number left under it has no perfect-cube factor other than 1. ∛2 and ∛4 are both finished forms — 4 cannot offer a cube — which is why 3∛4 stops there even though 4 itself is not prime.

Some radicands are finished from the start, and recognizing them saves wasted factoring. √30 is finished because 30 = 2 × 3 × 5, with no prime appearing twice, so no pair exists to pull out. √98 is not finished: 98 = 49 × 2, and 49 is 7^2, so √98 = 7√2. √200 is a good mental check: 200 = 100 × 2, so √200 = 10√2 — and 10√2 squared is 100 × 2 = 200, which confirms the answer by squaring it back. That squaring-back check works for any simplified radical: square the outside number, multiply by the inside number, and you must land back on the original radicand.

One boundary is worth stating plainly. This calculator works with real roots of non-negative radicands: negative numbers under a square root have no real simplest form, and the calculator says so rather than inventing one. Cube roots of negatives do exist in real mathematics, but the tool’s scope is the non-negative case its steps describe, and negative radicands are refused with that explanation instead of a silent wrong answer.

Ratios with decimals, and why scaling first is honest

Decimal ratios look harder than whole-number ratios, but they are the same job with one extra first step: scale both sides by the same power of ten until the decimals disappear, then simplify exactly as before. The calculator shows that scaling step explicitly, because hiding it would make the GCD step look like magic. For 2.5 : 7.5, multiplying both sides by 10 gives 25 : 75, whose GCD is 25, so the simplest ratio is 1 : 3. Nothing about the comparison changed — both sides were multiplied by the same number, which is the one operation that never changes a ratio.

The scaling factor is decided by the messiest side: count the decimal places on each side and take the larger count. For 0.6 : 0.9, one decimal place on each side means multiplying by 10: 6 : 9, GCD 3, simplest form 2 : 3. For 1.5 : 2.25, the second side has two decimal places, so both sides are multiplied by 100: 150 : 225, GCD 75, simplest form 2 : 3. The same pair of numbers can therefore arrive at 2 : 3 by different scalings, and the calculator’s step list shows which scaling it used, so the working can be followed rather than trusted blindly.

Equivalent ratios are the same idea run in reverse, and the calculator’s equivalent table is a check on the simplification. If 24 : 36 simplifies to 2 : 3, then multiplying both sides of 2 : 3 by 2, 3, 5 and 10 must give 4 : 6, 6 : 9, 10 : 15 and 20 : 30 — and every one of those ratios is worth exactly the same as 2 : 3. A simplified ratio that cannot regenerate its original by a single multiplication is wrong, and the equivalent table makes that test visible at a glance.

Ratios also carry a units warning that fractions do not need. A ratio only makes sense when both sides are measured in the same unit, so a comparison like 1 hour : 90 minutes must be converted first — 60 minutes : 90 minutes — before it simplifies to 2 : 3. The calculator simplifies the numbers you give it; it does not convert units for you, and feeding it mixed units will produce a correctly simplified but meaningless ratio. Convert first, simplify second, and the answer describes the real comparison.

Finally, a ratio and a fraction are close cousins but not twins, and the page keeps them apart on purpose. A fraction compares a part to a whole — 2/3 of a pizza is 2 pieces out of 3 in total — while a ratio compares a part to another part — 2 : 3 can mean 2 of one thing for every 3 of another, with 5 parts in total. The same two numbers therefore answer different questions depending on which form they wear, which is why the calculator reports both forms for ratio mode: the simplest ratio, and the same comparison written as a fraction.

Algebraic simplification: collecting like terms

The Advanced panel on this page adds a fourth job to the calculator’s three numeric modes: simplifying an algebraic expression in one variable, x, by collecting like terms. It is a separate mode on purpose. It shares no outputs with the fraction, radical and ratio modes, it never changes their results, and it runs only when you press its own Simplify button. If you never open the panel, the three numeric modes behave exactly as they always did.

Like terms are terms with exactly the same variable part. In 3x^2 + 2x – 5 + 4x^2 – x, the like pairs are 3x^2 and 4x^2 (both x^2), 2x and x (both x), and the constant -5, which has no variable part and therefore joins nothing. Adding each group gives 7x^2 + x – 5: the x^2 coefficients add to 7, the x coefficients add to 1 — written as x, not 1x — and the constant stays -5. The order of the answer follows the usual convention of highest power first, then lower powers, then the constant, so the result reads the way a textbook would write it.

Multiplication inside a term adds exponents, because x^2 × x^3 is five copies of x multiplied together. The panel applies that rule wherever terms are multiplied: 2x × 3x = 6x^2, since 2 × 3 = 6 and the exponents 1 + 1 = 2; and x^2 × x^3 + x^5 = 2x^5, since the product is x^5 and there is already an x^5 term waiting to join it. A term that cancels completely disappears from the answer — 2x^2 – 2x^2 leaves nothing in the x^2 group — and an expression that cancels entirely simplifies to 0, which the panel reports as 0 rather than as an empty box.

The panel’s scope is deliberately narrow, and its limits are part of its honesty. It handles one variable, x, with non-negative whole-number exponents, joined by plus, minus, multiplication and powers. It does not expand brackets, so 3(x + 2) is refused with an explanation instead of a guessed answer. It does not divide one expression by another, solve equations, or factor — 3x^2 + 2x – 5 is a simplified expression, not an equation to solve, and no value of x is ever claimed. Expressions with other letters, decimal exponents, or division are refused the same way, with the reason shown, because a simplifier that silently mishandles an input is worse than one that declines it.

A good check on any algebraic answer is substitution: pick a value for x, evaluate the original expression and the simplified expression, and the two numbers must match. For 3x^2 + 2x – 5 + 4x^2 – x at x = 2, the original is 12 + 4 – 5 + 16 – 2 = 25, and the simplified 7x^2 + x – 5 is 28 + 2 – 5 = 25. If a substitution ever disagrees, the simplification is wrong — and that test, unlike the algebra itself, needs no theory at all.

Five ways simplifying goes wrong

Most simplifying errors are not arithmetic slips; they are one of a small set of tempting moves that look like simplifying but change the value. Each one below is a real error with a small example you can check in seconds, and each one has a one-line rule that prevents it.

First, cancelling across a plus sign. (4 + 8) ÷ 4 is 3, because the division applies to the whole sum, 12 ÷ 4. Cancelling the 4 against the first term only gives 1 + 8 = 9, which is a different number wearing a plausible costume. The rule: only cancel a factor of the entire numerator or the entire denominator, never one term of a sum. If a sum must be cancelled, factor it first — (4 + 8) is 4 × 3, and then the 4 genuinely cancels.

Second, dividing only one side. Reducing 24/36 by dividing only the top by 12 gives 2/36, which is a twelfth of the original value, not an equivalent form. The rule: whatever divides the top must divide the bottom, and vice versa, because a fraction’s value is the relationship between its two parts. The calculator’s steps always show both divisions side by side for exactly this reason.

Third, stopping early. Dividing 24/36 by 6 gives 4/6, which is correct but unfinished — 4 and 6 still share 2. The rule: the GCD is the largest common divisor, and the finished test is that the simplified parts share no factor except 1. Dividing by any common factor is progress, but only dividing by the GCD finishes the job in one move.

Fourth, splitting a root over a sum. √(16 + 9) is √25, which is 5 — not √16 + √9, which is 4 + 3 = 7. Roots split over multiplication, never over addition: √(36 × 2) is √36 × √2, but √(36 + 2) cannot be split at all. The rule: simplify inside the root first, then pull factors out; never distribute a root across a plus sign.

Fifth, scaling only one side of a ratio. Doubling the left side of 2 : 3 gives 4 : 3, a ratio that favours the first quantity twice as much as before. The rule: a ratio survives only multiplications applied to both sides equally, which is why the equivalent table always multiplies both sides by the same number. The same caution applies to decimals: scaling 2.5 : 7.5 means multiplying both sides by 10, not just the side with the decimal point.

Underneath all five errors sits one question worth asking before any simplifying move: does this change the value? Simplifying, done right, never does — 24/36, 2/3, 0.666667 and 66.6667% are four costumes for one quantity. Any step that produces a different value has stopped being simplification and started being a different problem.

Simplifying questions, answered

The ideas underneath the buttons — GCDs, mixed numbers, radicals and the ratio–fraction line.

The wider fraction and roots toolkit

Everything around this page’s three modes, one click away.