Simplify Square Root Calculator: Simplest Radical Form

Q: How do you simplify √72?

Write 72 = 36 × 2 , where 36 is the largest perfect square that divides 72. Then √ 72 = √ 36 √ 2 = 6 √ 2 ≈ 8.4853 . With primes: 72 = 2 × 2 × 2 × 3 × 3 , the pairs give 2 × 3 = 6 outside and one 2 stays inside.

To simplify a square root, find the largest perfect square that divides the number and take its root outside: √72 = √36 · 2 = 6√2. A root is in simplest radical form when no square factor other than 1 is left under the sign. This simplify square root calculator does it by prime factorization and shows every step.

How to use the radical simplifier

Type the number under the root, for example 72. It has to be a whole number.
Optionally type a number in front, for expressions such as 3√20. Leave it empty for a plain root.
Read the answer. You get the simplest radical form, the decimal rounded to the places you choose, the prime factorization, the largest square factor and what is left under the root.
Follow the factor tree. Each level splits off the smallest prime. Green primes come in pairs and move outside the root; orange primes are single and stay inside.

Why Pythagorean answers are so often square roots

Solving a right triangle ends with a square root: c = √a² + b². The answer is a whole number only when a² + b² happens to be a perfect square, as in the 3-4-5 triangle. Most of the time it is not. Legs of 2 and 4 give c = √20; legs of 5 and 5 give √50; a hypotenuse of 8 and a leg of 4 leave the missing leg as √48.

Teachers and textbooks usually want these answers in simplest radical form, 2√5, 5√2 and 4√3, because the form is exact and makes patterns visible. Every 45-45-90 triangle has a √2 in it and every 30-60-90 triangle has a √3. The decimal 7.0711 hides that; 5√2 shows it. The Pythagorean theorem calculator simplifies its answers automatically, and this page shows how. The spiral of Theodorus builds √2, √3, √4 and so on as the hypotenuses of a chain of right triangles.

Method 1: prime factors in pairs

This is the method the calculator uses, because it always works and always finishes in simplest form.

Factor the number into primes. 180 = 2 × 2 × 3 × 3 × 5. A factor tree is the easiest way to find them.
Group equal primes in pairs. 180 = (2 × 2) × (3 × 3) × 5.
Move one number out for each pair. A pair is a perfect square, and √p × p = p. So √180 = 2 × 3 × √5.
Multiply what is outside and what is inside. √180 = 6√5 ≈ 13.4164.

The rule behind every step is √xy = √x √y for positive numbers, so a square root can be split across a product and each square factor taken out separately.

Method 2: the largest perfect-square factor

If you know your perfect squares (4, 9, 16, 25, 36, 49, 64, 81, 100, …), you can often see the answer in one step. Look for the largest one that divides your number:

√200 = √100 · 2 = √100 · √2 = 10√2

Taking out the square factor 36 = 6² means counting how many small sides of length √2 fit along the big square: six.

Watch outIf you use a square factor that is not the largest, you are not finished. √72 = √4 · 18 = 2√18 is correct but not simplest, because 18 still contains the square 9. Keep going: 2√18 = 2 · 3√2 = 6√2.

Worked examples

Example 1: the hypotenuse of a 5-5 right triangle

The legs are both 5, so c = √25 + 25 = √50.

√50 = √25 · 2 = 5√2 ≈ 7.0711

√50 = 5√2. The factor tree is 50 = 2 × 5 × 5: one pair of 5s comes out, the single 2 stays in.

Example 2: a number in front of the root

Simplify 3√20.

3√20 = 3√4 · 5 = 3 · 2√5 = 6√5 ≈ 13.4164

3√20 = 6√5. Simplify the root first, then multiply by the number already outside.

Example 3: a missing leg

A right triangle has hypotenuse 8 and one leg 4. The other leg is √64 − 16 = √48.

√48 = √16 · 3 = 4√3 ≈ 6.9282

The leg is 4√3. The hypotenuse is twice the short leg, so this is a 30-60-90 triangle.

Example 4: a larger number

Simplify √1000.

1000 = 2³ × 5³ = (2 × 2) × (5 × 5) × 2 × 5 √1000 = 2 × 5 × √2 × 5 = 10√10 ≈ 31.6228

√1000 = 10√10. Leftover singles multiply together under the root.

Common square roots in right-triangle problems

√n	Simplest form	Decimal	Typical source
√8	2√2	2.8284	legs 2 and 2
√12	2√3	3.4641	hypotenuse 4, leg 2
√18	3√2	4.2426	legs 3 and 3
√20	2√5	4.4721	legs 2 and 4
√27	3√3	5.1962	hypotenuse 6, leg 3
√32	4√2	5.6569	legs 4 and 4
√45	3√5	6.7082	legs 3 and 6
√48	4√3	6.9282	hypotenuse 8, leg 4
√50	5√2	7.0711	legs 5 and 5, or 1 and 7
√52	2√13	7.2111	legs 4 and 6
√72	6√2	8.4853	legs 6 and 6
√75	5√3	8.6603	hypotenuse 10, leg 5
√80	4√5	8.9443	legs 4 and 8
√98	7√2	9.8995	legs 7 and 7
√200	10√2	14.1421	legs 10 and 10

Decimals are rounded to four places. Roots of numbers with no square factor, such as √13, √29 or √30, are already in simplest form; the square roots and radicals lesson has a full list of perfect squares and more practice.

Check your answer by estimating

A quick estimate catches most slips. Find the two perfect squares on either side of your number. Since 50 lies between 49 and 64, √50 lies between 7 and 8, and because 50 is much closer to 49 it should be just over 7. If you had simplified √50 to 2√5 ≈ 4.47, the estimate would show at once that something went wrong.

50 is just above the perfect square 49, so √50 is just above 7.

Rationalizing the denominator

Simplest radical form also means no square root left in the bottom of a fraction. To remove one, multiply the top and bottom by that root:

6 over √3 = 6√3 over 3 = 2√3 10 over √2 = 10√2 over 2 = 5√2

This step appears constantly with special triangles, for example when you find the leg of a 45-45-90 triangle from its hypotenuse. The special right triangle calculator rationalizes automatically and shows the step.

Common mistakes

Splitting a sum. √9 + 16 = √25 = 5, not √9 + √16 = 7. Roots split across products, never across sums. This is the classic error described on the common mistakes page.
Taking out a single prime. In √12 = √2 × 2 × 3 only the pair of 2s comes out, giving 2√3. The 3 cannot come out on its own.
Forgetting the number already in front. 3√20 is 3 × 2√5 = 6√5, not 2√5.
Adding unlike radicals. √2 + √3 is not √5. Only roots with the same number inside combine: 2√5 + 3√5 = 5√5.

Frequently asked questions

What is simplest radical form?

A square root is in simplest radical form when the number under the root has no perfect-square factor except 1, and no root is left in a denominator. 6√2 is simplest; √72 and 2√18 are not.

How do you simplify √72?

Write 72 = 36 × 2, where 36 is the largest perfect square that divides 72. Then √72 = √36 √2 = 6√2 ≈ 8.4853. With primes: 72 = 2 × 2 × 2 × 3 × 3, the pairs give 2 × 3 = 6 outside and one 2 stays inside.

Can every square root be simplified?

No. If the number has no repeated prime factor, as with 2, 6, 15 or 30, its square root is already in simplest form. Prime numbers are the simplest case: √7 cannot be simplified.

Why not just use the decimal?

Because the decimal of an irrational root never ends, so any decimal is rounded and slightly wrong. Rounding early in a multi-step problem lets the error grow. Keep the exact form until the last step, then round once. The reasons √2 can never be written as a fraction are on the square root of 2 is irrational page.

How do I simplify a square root without a calculator?

Divide by small primes (2, 3, 5, 7, …) until you are left with 1, writing down each prime. That is the factor tree. Then pair up equal primes: each pair puts one copy outside the root and the singles stay inside.

Does this work for cube roots?

The same idea works, but you take out groups of three equal primes instead of pairs: ³√54 = ³√27 · 2 = 3³√2. This calculator handles square roots only.

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