Table of Contents
- What Is a Square Root?
- How to Use the Square Root Calculator
- Worked Example: The Square Root of 72 Step by Step
- Perfect Squares and How to Spot Them
- Square Root Examples
- Where Square Roots Show Up in Real Life
- Common Mistakes to Avoid
- FAQs
The square root of a number answers a simple question: what number, multiplied by itself, gives this number back? Sometimes the answer is neat, like the square root of 144, which is exactly 12. More often it is a long decimal that needs simplifying before it makes sense, like the square root of 72. Use the calculator at the top of this page to get any square root instantly, with the decimal value, the simplified radical form, and the working shown step by step. This guide explains what a square root really is, how the simplification works, and how to spot the answers that come out exact.
Square Root Calculator
| Square root (4 decimal places) | |
| Square root (2 decimal places) | |
| Simplified radical form | |
| Perfect square? | |
| Prime factorisation of your number |
Whole numbers up to 1,000,000,000 get the full step-by-step working. Decimals get the root value.
What Is a Square Root?
Squaring a number means multiplying it by itself: 5 squared is 5 × 5 = 25. A square root undoes that. The square root of 25 is 5, because 5 × 5 = 25. The symbol for this is √, called the radical sign, and the number under it is called the radicand. As Wolfram MathWorld’s square root page puts it, a square root of a number is any value that gives the number back when multiplied by itself.
Every positive number actually has two square roots, a positive one and a negative one, because −5 × −5 = 25 as well. The one we use in everyday maths is the positive one, known as the principal square root, and that is what this calculator shows. Zero has one square root, itself, and negative numbers have no square root in the real numbers at all — a point we come back to in the FAQs.
Square roots also connect to powers. Taking a square root is the same as raising a number to the power of one half, so √25 can also be written as 25^(1/2). This is why roots appear so often next to squares and square numbers in algebra, geometry and science.
How to Use the Square Root Calculator
Type any number into the calculator at the top of this page and press Calculate. You will see the square root to four decimal places, the same value rounded to two decimal places, and, for whole numbers, a set of extra results that most calculators skip.
First, the tool checks whether your number is a perfect square. If it is, you get the exact whole-number answer. Next, it shows the simplified radical form, which rewrites the root in its tidiest exact form — for example, √72 becomes 6√2. Finally, it lists the prime factorisation of your number and a short set of steps explaining how the simplified form was found. The result updates as you type, so you can try nearby numbers and watch the pattern change.
Whole numbers up to 1,000,000,000 get the full step-by-step treatment. Decimals work too — enter 2.25, for instance, and you get 1.5 — but the simplifying steps are written for whole numbers, because that is where they are useful.
Worked Example: The Square Root of 72 Step by Step
Seventy-two is a good example because it is not a perfect square, yet it simplifies nicely. Here is the method the calculator uses, which is the same method taught in school.
Step 1: factorise the number. Break 72 into prime factors: 72 = 2 × 2 × 2 × 3 × 3. You can find this by dividing by the smallest prime that fits, again and again: 72 ÷ 2 = 36, 36 ÷ 2 = 18, 18 ÷ 2 = 9, and 9 ÷ 3 = 3, then 3 ÷ 3 = 1.
Step 2: pair up the factors. Square roots work in pairs, because a pair of identical factors multiplied together makes a perfect square. In 2 × 2 × 2 × 3 × 3 there is one pair of 2s, one pair of 3s, and one spare 2 left over.
Step 3: bring one of each pair outside. Each pair contributes a single factor outside the root sign: 2 × 3 = 6. The spare 2 stays inside. So √72 = 6√2.
Checking the decimal value
Step 4: check the decimal. √2 is about 1.4142, so 6√2 ≈ 6 × 1.4142 = 8.4853. Rounded to two decimal places, the square root of 72 is 8.49. There is a shortcut version of the same idea: find the largest perfect square that divides your number (36 divides 72), write √72 = √(36 × 2) = √36 × √2 = 6√2. Both routes land in the same place. Wolfram MathWorld’s page on Newton’s iteration describes the classic method computers use to home in on the decimal value itself.
| Step | Working | Result |
|---|---|---|
| Prime factorisation | 72 = 2 × 2 × 2 × 3 × 3 | Pairs: (2, 2), (3, 3), spare 2 |
| Take one from each pair | 2 × 3 | 6 outside the root |
| Leftover stays inside | Spare factor 2 | √2 stays as a root |
| Simplified form | 6 × √2 | 6√2 |
| Decimal value | 6 × 1.4142 | 8.4853 (8.49 to 2dp) |
Perfect Squares and How to Spot Them
A perfect square is a whole number whose square root is also a whole number. The first few are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144. Learning to recognise them pays off twice: you get instant exact answers, and they are the building blocks for simplifying other roots, as the 72 example showed.
| Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Squared | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
A quick spotting trick: perfect squares never end in 2, 3, 7 or 8. If a number ends in one of those digits, its root cannot be a whole number, so you can go straight to simplifying or to the decimal. Squares of numbers ending in 5 are easy too — 25, 225, 625 — they always end in 25.
Square Root Examples
The table below collects the cases people ask about most, all produced by the same method as the worked example. Notice how 50 and 200 simplify to a whole number times √2, because both hide a perfect square times 2 inside them.
| Number | Simplified form | Decimal (2dp) | Perfect square? |
|---|---|---|---|
| 2 | √2 | 1.41 | No |
| 12 | 2√3 | 3.46 | No |
| 50 | 5√2 | 7.07 | No |
| 72 | 6√2 | 8.49 | No |
| 144 | 12 | 12.00 | Yes |
| 200 | 10√2 | 14.14 | No |
Where Square Roots Show Up in Real Life
The most common everyday use is Pythagoras’ theorem. If you know the two shorter sides of a right-angled triangle, the longest side is the square root of the sum of their squares — the same maths our area of a triangle calculator sits alongside in geometry work. A 3–4–5 triangle is the famous case: √(9 + 16) = √25 = 5.
Square roots also run in the other direction through area. A square garden with an area of 144 square metres has sides of √144 = 12 metres, which is the quickest way to turn a paving or fencing quote based on area into real lengths. In statistics, the standard deviation is a square root — it is the root of the average squared difference from the mean — and in finance, volatility figures come from the same idea. Even simplifying roots by hand leans on skills from other topics: the pairing method is really factorising, which is why it pairs naturally with our greatest common factor calculator when you are breaking numbers down.
Common Mistakes to Avoid
The classic error is assuming a root can be split over addition: √(9 + 16) is 5, but √9 + √16 is 3 + 4 = 7. Roots split over multiplication, never over addition or subtraction. A second mistake is rounding too early — if you round √2 to 1.41 and then multiply by 6, you get 8.46 instead of 8.49, a small error that grows in longer calculations. Keep the exact form, 6√2, until the final step.
People also forget the negative root. The equation x² = 25 has two solutions, 5 and −5, even though the calculator shows the principal (positive) root. Finally, when simplifying, check that the number left inside the root has no square factors left: √200 is not finished at 5√8, because 8 still hides a factor of 4. The finished answer is 10√2. If your simplified form still contains an even power or a factor like 4, 9 or 25 under the sign, there is more to bring out. Roots of fractions follow their own neat rule — take the root of the top and the root of the bottom separately — which our fraction calculator can help you tidy up afterwards.
FAQs
What is the square root of 72 in simplest form?
The square root of 72 simplifies to 6√2. Factorising 72 gives 2 × 2 × 2 × 3 × 3; one 2 and one 3 come out of each pair to make 6, and a single 2 is left inside the root. As a decimal, √72 ≈ 8.4853, or 8.49 to two decimal places.
How do you find a square root without a calculator?
Start by finding two perfect squares your number sits between. Since 72 lies between 64 and 81, its root lies between 8 and 9. Then refine: 8.5² = 72.25, which is slightly too big, and 8.49² ≈ 72.08, so the root is about 8.485. For an exact simplified answer, factorise the number and pair up the prime factors instead.
What is a perfect square?
A perfect square is a whole number that is the square of another whole number, so its square root comes out exact. Examples are 36 (6 × 6), 100 (10 × 10) and 144 (12 × 12). The calculator flags perfect squares for you and gives the whole-number root.
Can a negative number have a square root?
Not in the real numbers, because a number multiplied by itself is never negative. Roots of negative numbers are handled with the imaginary unit i, defined so that i² = −1; for example, the square root of −9 is written as 3i. This calculator works with real numbers, so it explains this instead of showing a real result for a negative input.
What is the square root of 2?
The square root of 2 is approximately 1.4142. It cannot be simplified and cannot be written exactly as a decimal or a fraction, because it is an irrational number — MathWorld notes it is sometimes called Pythagoras’s constant. It turns up constantly, for instance as the diagonal of a square with sides of length 1.
How do you find the square root of a fraction?
Take the square root of the numerator and the square root of the denominator separately. For example, √(9/25) = √9 ÷ √25 = 3/5 = 0.6. If the two parts do not come out exact, simplify the fraction first and then treat the top and bottom in the same way.
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