What does it mean to simplify a radical?

To move every square factor out from under the root and in front of it as a coefficient. √72 becomes 6√2 because 72 = 36 × 2 and 36 is a perfect square. The value does not change — 6√2 is 8.485281 and so is √72 — but the form is the one a further calculation can use.

When are two radicals like terms?

When the numbers under the roots are equal after simplifying. 6√2 and 3√2 are like terms and add to 9√2; √2 and √3 are not, and their sum cannot be written as a single radical. Only the coefficients are added — the radicand is the unit being counted, so it stays put.

Why is the answer given as an exact form rather than a decimal?

Because most radicals are irrational, and a decimal can only ever approximate them. 7√2 is the exact value; 9.899495 is the same number rounded to six places and is wrong in the seventh. Keeping the exact form is what lets the answer be used in a later step without the rounding piling up.

Can I enter a cube root?

Not on this page. ∛8 + ∛2 combines by exactly the same rule as the square roots do, but the notation needs an index written into the radical sign, and this page prints a bare √. Use root-calculator for one term at a time with any index from 2 to 12.

Why is a whole number allowed alongside the radicals?

Because a whole number is a radical whose radicand has come down to 1 — √25 is 5, and 5 is 5√1. Treating it that way means whole numbers and radicals combine by one rule, and gives the fixed order the answer is printed in: 5 + √2, with the whole number first.

How do I check the answer?

Square the coefficient and multiply it back under the root. 7√2 checks as 49 × 2 = 98, which is not the original sum — so check it term by term instead: 6√2 is 36 × 2 = 72, 3√2 is 18, and 2√2 is 8, and 72 + 18 - 8 = 82. That is the sum of the radicands as written, which is why the check has to be done before the terms are combined.

Radical Calculator

Result

7√2

Exact form

As a decimal: 9.899495

Simplify radicals by hand and see why two terms can be added at all: this page pulls the largest square factor out of every term in a sum, then combines the terms whose radicand — the number left under the root — matches. √72 + 3√2 - √8 becomes 6√2 + 3√2 - 2√2, and because all three are now multiples of √2 they add up to 7√2. Simplest radical form is the stopping point: nothing left inside a root can be squared out, and no two terms can be merged. √2 + √3 is already there.

√72 + 3√2 - √8, one term at a time

Term	Coefficient	Radicand	Coefficient after simplifying	Radicand after simplifying
√72	1	72	6	2
3√2	3	2	3	2
-√8	-1	8	-2	2

One row per term, in the order they were written. The first three columns are the term as it arrived; the last two are what it becomes once the largest square factor has been pulled out. Reading the last column downwards is the whole point of the table: all three terms end at 2, which is the discovery that lets the coefficients on the fourth column be added — 6 + 3 - 2 = 7. A term whose radicand is already square-free, like 3√2, passes through unchanged. Everything here is a whole number, so the table is identical in all ten languages the site serves.

Formula

c√r → (c × m)√(r ÷ m²), where m² is the largest square dividing r like terms: k₁√r + k₂√r = (k₁ + k₂)√r

√r: The radical: the square root of a whole number r. Every term on this page is a multiple of one of these, and two terms are alike exactly when their r matches — the coefficient in front is what gets added.
c: The coefficient: the whole number multiplying the radical, 1 when nothing is written. It counts how many copies of √r are being added, so 3√2 plus 2√2 is five of them, and 3√2 minus 3√2 leaves none.
r: The radicand: the number under the root. Simplifying a single radical means finding the largest square m² that divides r and moving m outside, which leaves a smaller radicand behind. The smallest radicand a term can reach is 1, and √1 is just 1.
m²: The largest square dividing the radicand — 36 for 72, 25 for 50, 4 for 8. Taking the largest one is what makes the answer the simplest radical form rather than merely a smaller one: 72 divided by 4 gives 18, and 18 still has a square factor left in it.
like terms: Terms whose radicands match after simplifying. Adding radicals is only ever adding like terms: the radicand is the unit being counted, and it never changes when two of them are added. That is why combining 6√2 and 3√2 gives 9√2 rather than 9√4.

Use this when a radical expression — a sum of square roots — has to be written in its simplest radical form: checking homework, tidying an answer before it goes into a formula, or seeing whether two radicals that look different are in fact the same number underneath. The check that goes with it is to square the coefficient and multiply it back inside: 6√2 is 36 × 2 = 72, so 6√2 and √72 really are the same.

Worked examples

√72 + 3√2 - √8
1. √72: 72 = 36 × 2, so the largest square factor 36 comes out as 6, leaving 6√2
2. 3√2: the radicand 2 has no square factor, so this term is already in simplest form
3. √8: 8 = 4 × 2, so 4 comes out as 2, leaving -2√2
4. The three terms are now 6√2 + 3√2 - 2√2 — all multiples of √2, so the radicand stays and the coefficients add
5. 6 + 3 - 2 = 7, so the sum is 7√2
6. 7√2 is 9.899495 to six places
The default case, and the reason this page exists rather than three separate simplifications: the three radicals start out looking unrelated (72, 2 and 8) and turn out to be the same radical counted 6, 3 and 2 times. The radicand never changes while they are added — only the count in front of it does.
√50 + 5 - √18
1. √50: 50 = 25 × 2, so 25 comes out as 5, leaving 5√2
2. √18: 18 = 9 × 2, so 9 comes out as 3, leaving -3√2
3. 5 is already a whole number — there is nothing under a root to simplify
4. The radical terms are 5√2 - 3√2 = 2√2
5. The whole number stays where it is, so the answer is 5 + 2√2
6. 5 + 2√2 is 7.828427 to six places
A whole number and a radical are not like terms, so they sit side by side rather than being added into one figure. They are still printed in a fixed order, whole number first, which is what makes 5 + 2√2 the answer here and 2√2 + 5 the answer to the same sum written the other way round.
√2 + √3, where nothing combines
1. √2: the radicand 2 has no square factor other than 1, so nothing comes out
2. √3: 3 has no square factor either, so nothing comes out
3. The two radicands are 2 and 3 — not equal, so the terms are not like terms
4. The sum is written as it stands, √2 + √3
5. √2 + √3 is 3.146264 to six places
Nothing is done here, and that is the answer: radicals whose radicands differ cannot be added into a single radical, because there is no number whose square is 5 that could stand in for √2 + √3. The decimal is still given, since the value is perfectly well defined — it just has no shorter exact form.

Limitations

Square roots only. Cube roots and higher combine by exactly the same rule — the radicands are compared and the coefficients added — but writing them needs a symbol that carries the index, and this page's notation has no room for one; root-calculator handles the higher roots, one term at a time. Only whole numbers are accepted, in the coefficient and under the root alike: 2.5√2 has an exact form of (5√2)/2, which needs a fraction, and √2.5 is √10 ÷ 2 for the same reason. Negative radicands are refused because no real number squares to a negative one. Up to eight terms are taken, and each coefficient and radicand is capped at one million. Decimals in the answer are rounded to six places, so they are approximations of an exact value rather than the value itself.

Frequently asked questions

What does it mean to simplify a radical?: To move every square factor out from under the root and in front of it as a coefficient. √72 becomes 6√2 because 72 = 36 × 2 and 36 is a perfect square. The value does not change — 6√2 is 8.485281 and so is √72 — but the form is the one a further calculation can use.
When are two radicals like terms?: When the numbers under the roots are equal after simplifying. 6√2 and 3√2 are like terms and add to 9√2; √2 and √3 are not, and their sum cannot be written as a single radical. Only the coefficients are added — the radicand is the unit being counted, so it stays put.
Why is the answer given as an exact form rather than a decimal?: Because most radicals are irrational, and a decimal can only ever approximate them. 7√2 is the exact value; 9.899495 is the same number rounded to six places and is wrong in the seventh. Keeping the exact form is what lets the answer be used in a later step without the rounding piling up.
Can I enter a cube root?: Not on this page. ∛8 + ∛2 combines by exactly the same rule as the square roots do, but the notation needs an index written into the radical sign, and this page prints a bare √. Use root-calculator for one term at a time with any index from 2 to 12.
Why is a whole number allowed alongside the radicals?: Because a whole number is a radical whose radicand has come down to 1 — √25 is 5, and 5 is 5√1. Treating it that way means whole numbers and radicals combine by one rule, and gives the fixed order the answer is printed in: 5 + √2, with the whole number first.
How do I check the answer?: Square the coefficient and multiply it back under the root. 7√2 checks as 49 × 2 = 98, which is not the original sum — so check it term by term instead: 6√2 is 36 × 2 = 72, 3√2 is 18, and 2√2 is 8, and 72 + 18 - 8 = 82. That is the sum of the radicands as written, which is why the check has to be done before the terms are combined.

References

Quadratic Surd — the canonical form a√b with b square-free, which is exactly what the simplified radicand column is — Wolfram MathWorld (United States)
Radical — the radical sign and the operation it names, including why two radicals can only be added when their radicands agree — Wolfram MathWorld (United States)
Like terms — what makes two terms addable, which for radicals is decided by the number under the root — SplashLearn (United States)

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