Square Root Calculator
MathEnter a number and choose how many decimal places to show. You get the square root, whether the number is a perfect square, the two nearest perfect squares and the simplified radical, with the steps.
Step by step
√50 ≈ 7.0711
- The square root of 50 is the non-negative number that gives 50 when multiplied by itself.
- The whole-number part is 7: 7² = 49 does not go above 50, while the next whole number squared, 64, does.
- Rounded to 4 decimal places, the result is 7.0711.
Once every entry is valid, you’ll see each step of the calculation here.
Formula
- x
- the radicand, the number you want the root of, zero or more
- y
- the square root, the non-negative number that gives x when multiplied by itself
- s, q
- the part that moves outside the root and the part that stays, with no factor that appears twice
- yn
- the estimate at step n, which gets closer to the root with each step
How it works
Enter the number you want the root of, called the radicand, and choose how many decimal places you want in the result. The calculator finds the principal square root, the non-negative number that gives the radicand when multiplied by itself, and checks whether the radicand is a perfect square. It works with whole numbers and decimals from 0 to one trillion.
To estimate a root by hand, learn the first perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361 and 400. For √50, notice that 49 is less than 50 and 64 is more, so the root lies between 7 and 8, just above 7. The calculator shows the whole-number part and the two nearest perfect squares for that reason: they are the first estimate for any root. To sharpen an estimate, average it with the number divided by it. Starting from 7, (7 + 50 / 7) / 2 = 7.0714, and the next round gives 7.07107.
If the radicand is a whole number, you can also factor it into primes. Every factor that appears twice moves outside the root: 72 = 2³ × 3² = 2² × 3² × 2, so √72 = 2 × 3 × √2 = 6√2. When every exponent is even the number is a perfect square and the root is a whole number: 144 = 2⁴ × 3², so √144 = 2² × 3 = 12. Whole numbers with a square factor also get a simplified radical in the result.
Square roots turn up whenever an area is known and a side is needed. A square patio of 400 square feet has sides of 20 feet. For areas, see the square footage calculator. The triangle calculator uses a square root in the Pythagorean theorem, the circle calculator finds a radius from an area, and the quadratic formula calculator takes the root of the discriminant.
The formula
The square root undoes squaring: √144 = 12 because 12 × 12 = 144. The symbol √ always means the principal root, the non-negative one. Both 12 and −12 give 144 when squared, but √144 is 12. The square of a real number is never negative, so among the real numbers a square root exists only for numbers that are zero or more.
When the radicand is a whole number that is not a perfect square, its root is irrational. It has endless decimals that never repeat in a pattern, like √2 = 1.41421356… The calculator rounds it to the number of decimal places you choose, from 2 to 8. For a whole number it also writes the simplified radical, so 50 = 25 × 2 gives √50 = 5√2.
The averaging method, known as the Babylonian or Heron’s method, gives a better estimate at each step: take the average of the current estimate y and x / y. For √50 start at 7. The first step gives 7.0714, and the next gives 7.07107, which is the true value 7.0710678… rounded to five decimals.
Example
What is the square root of 50?
- The nearest perfect squares are 49 and 64, so the square root of 50 lies between 7 and the next whole number.
- Factor the radicand: 50 = 25 × 2 = 5² × 2, so the simplified radical is 5√2.
- Rounded to 4 decimal places the root is 7.0711. The averaging method agrees: (7 + 50 / 7) / 2 = 7.0714, then 7.07107.
The square root of 50 is about 7.0711, rounded to 4 decimal places. 50 is not a perfect square, so its root is irrational.
Load this example into the calculatorFrequently asked questions
What is a square root?
The square root of a number x is the non-negative number that gives x when you multiply it by itself. It is written √x, so √81 = 9 because 9 × 9 = 81. It is the reverse of squaring, and among the real numbers it exists only when x is zero or more.
How do I calculate a square root?
On a calculator, enter the number and press the √ key. Without one, find the two perfect squares around the number, which gives a whole-number estimate, then refine it by averaging it with the number divided by it. For an exact value, factor the number into primes and move each pair of equal factors outside the root.
How do I calculate the square root of a number by hand?
Estimate, then refine. For √50, 7² = 49 and 8² = 64, so the root is just over 7. Average 7 with 50 / 7 to get 7.0714, then average that with 50 / 7.0714 to get 7.07107. Each round roughly doubles the number of correct digits. The calculator above shows the bounds and the answer.
What is the square root of 2, 3 or 5?
They are irrational, so the decimals never end. √2 is 1.41421356, √3 is 1.73205081 and √5 is 2.23606798, rounded to eight decimal places. Enter 2, 3 or 5 and choose 8 decimal places to see them. None of the three can be simplified, because 2, 3 and 5 are prime.
How do I simplify a square root or reduce a radical?
Factor the number into primes and move each pair of equal factors outside the root. √72 = √(2² × 3² × 2) = 2 × 3 × √2 = 6√2. Another way is to pull out the largest perfect square that divides the number: 50 = 25 × 2, so √50 = 5√2. The calculator shows the simplified radical for whole numbers.
What is a perfect square and how can I tell?
A perfect square is a whole number that is the square of another whole number: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on. Its square root is a whole number. In its prime factorization every exponent is even. A quick check is the last digit, since a perfect square ends only in 0, 1, 4, 5, 6 or 9. The reverse does not hold: 21 ends in 1 but is not a perfect square.
Can you take the square root of a negative number?
Not among the real numbers, because squaring never gives a negative result. That is why the calculator asks for 0 or a positive number. In the complex numbers, the imaginary unit i is defined by i × i = −1, and √(−4) = 2i.
Does a positive number have two square roots?
Yes. 3 × 3 = 9 and (−3) × (−3) = 9, so 9 has the roots 3 and −3. The symbol √ means the principal, positive root, so √9 = 3. For the negative root, put a minus sign in front of the result: −√9 = −3.
How do I type the square root symbol?
You can copy √ from this page. On a Mac, press Option + V. On a Windows PC, hold Alt and type 251 on the numeric keypad, or type 221A in Word and press Alt + X. On a phone, look in the symbols or math keyboard. The Unicode name is "square root", code point U+221A.
How do I find a square root on a calculator with a square root button?
On most calculators, press the √ key and then the number, or the number and then √, depending on the model. On a scientific calculator, the key may be labeled √x or sqrt. You can also raise a number to the power 0.5: 50^0.5 = 7.0710678. This page does the same without a device and adds the steps.
Why use this square root calculator instead of a basic calculator?
A basic calculator gives one decimal value. This page also shows whether the number is a perfect square, the whole-number part, the two nearest perfect squares, the simplified radical such as 5√2 and the steps. You choose the number of decimal places, from 2 to 8.
What is the square root of a decimal or a fraction?
The calculator accepts decimals, like 2.25, and gives √2.25 = 1.5. For a fraction, take the root of the top and the bottom separately: √(9/16) = 3/4 = 0.75. If the fraction is not made of perfect squares, divide it into a decimal first and enter that. √(1/2) = √0.5 = 0.70710678.
How large a number can I enter?
Any number from 0 to 1,000,000,000,000 (one trillion), whole or decimal. The limit exists because the calculation uses double precision, which holds about 15 significant digits. For a number with more digits than that, the result would no longer be reliable to the last decimal.
- Common Core State Standards, Grade 8 Expressions and Equations (8.EE.A.2): square root symbols, perfect squares, irrational numbers (Common Core State Standards Initiative)
- Square root: definition, principal root and irrational roots (Wikipedia)
- Methods of computing square roots: the Babylonian method (Wikipedia)
- Square number: perfect squares and their last digits (Wikipedia)