What is the square root calculator?
This square root calculator finds the square root of any number in an instant and shows how the answer is reached. Enter a number and it returns the decimal value, the simplified radical form (for example √72 = 6√2), whether the number is a perfect square, and the prime factorization steps. Switch the Root option to get a cube root or any nth root. It is a free, online square root calculator with steps that works on phones, so you do not need a scientific calculator.
What is a square root?
The square root of a number x is the value that, multiplied by itself, gives x. Because 7 × 7 = 49, the square root of 49 is 7, written √49 = 7. The symbol √ is called the radical sign and the number under it is the radicand. Every positive number has two square roots, one positive and one negative (7 and −7), but √ by convention means the positive one, called the principal square root. The square root is the inverse of squaring: squaring 7 gives 49, and taking the square root of 49 gives back 7.
How to use this calculator
- Type the number. Decimals, fractions written as decimals (0.25) and large numbers all work.
- Choose Square root, Cube root or Nth root. For nth root, enter the index n (4 for the fourth root, 5 for the fifth, and so on).
- Read the decimal answer, then the simplified form and the working steps below it.
Square root formula
√x = x1/2 · ∛x = x1/3 · ⁿ√x = x1/n
A root is simply a fractional exponent. This means you can also find any root with an exponent calculator by raising the number to the power 1/n, or with logarithms: ⁿ√x = 10(log x) ÷ n (see the log calculator).
How to simplify a square root
To simplify a radical, find the largest perfect-square factor of the number and take its root outside the radical sign. The prime factorization method makes this systematic:
- Write the number as a product of primes: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3².
- Group equal primes in pairs (for a square root): (2 × 2), (3 × 3) and one leftover 2.
- Take one number out of each pair: 2 × 3 = 6. The leftover 2 stays inside.
- Result: √72 = 6√2 ≈ 8.4853.
For a cube root, group primes in threes; for an nth root, in groups of n. For example ∛54 = ∛(2 × 3³) = 3∛2. Our prime factorization calculator shows the prime factors of any number.
Square roots of common perfect squares
| Number | √ | Number | √ | Number | √ |
|---|---|---|---|---|---|
| 1 | 1 | 81 | 9 | 289 | 17 |
| 4 | 2 | 100 | 10 | 324 | 18 |
| 9 | 3 | 121 | 11 | 361 | 19 |
| 16 | 4 | 144 | 12 | 400 | 20 |
| 25 | 5 | 169 | 13 | 441 | 21 |
| 36 | 6 | 196 | 14 | 484 | 22 |
| 49 | 7 | 225 | 15 | 529 | 23 |
| 64 | 8 | 256 | 16 | 625 | 25 |
Useful non-perfect values: √2 ≈ 1.41421, √3 ≈ 1.73205, √5 ≈ 2.23607, √10 ≈ 3.16228.
What is the square root of 2?
√2 ≈ 1.41421356. It is an irrational number: its decimals never end or repeat, so it cannot be written as a fraction. It is the length of the diagonal of a square with sides of 1, which is why it shows up with the Pythagorean theorem calculator.
How to calculate square root manually
Long division method
The long division method, taught in Indian schools in class 8, finds the square root digit by digit by hand. Example: √529.
- Pair the digits from the decimal point: 5 | 29.
- Find the largest square ≤ 5: 2² = 4. Write 2 as the first digit, subtract: 5 − 4 = 1.
- Bring down the next pair: 129. Double the root so far (2 → 4) and find digit d so that (40 + d) × d ≤ 129. d = 3 gives 43 × 3 = 129.
- Remainder 0, so √529 = 23.
For non-perfect squares, add pairs of zeros after the decimal point and keep going for as many decimal places as you need.
Estimation trick
To estimate √50 without a calculator, find the nearest perfect squares: 49 and 64. √50 is just above 7. A quick refinement: √x ≈ a + (x − a²) ÷ 2a, where a² is the nearest square. So √50 ≈ 7 + 1/14 ≈ 7.071, very close to the true 7.0711.
Babylonian (Newton’s) method
Guess g, then repeat g = (g + x ÷ g) ÷ 2. For √10 starting at 3: 3.1667, then 3.16228. Each round roughly doubles the correct digits; this is the kind of algorithm computers use to find square roots.
Square root of negative numbers and fractions
A negative number has no real square root, because any real number squared is positive. In complex numbers, √(−16) = 4i, where i = √(−1); the calculator shows this. Cube roots and other odd roots of negative numbers are real: ∛(−27) = −3. For fractions, take the root of the top and bottom separately: √(9/16) = 3/4 = 0.75. As a decimal, enter 0.5625 and you get 0.75.
How to do square root on a calculator
On a basic or phone calculator, type the number and press the √ key; on some models you press √ first. On an iPhone, turn the phone sideways to open the scientific calculator and use the 2nd key for cube roots. On a Casio or TI-84, use √( for square roots and the x√ key (or MATH menu) for other roots. On a simple calculator with no root key, use the estimation or Babylonian method above.
Square root calculator formula, symbol and index
The square root calculator formula is √x = x1/2: the number that, multiplied by itself, gives x. The square root symbol is √ (the radical sign). A small number written in the crook of the sign is the index: ∛x has index 3, ⁴√x has index 4. With the square root index calculator option (Nth root), you choose any index n and get x1/n.
Square root calculation rules
- √(a × b) = √a × √b, so √72 = √36 × √2 = 6√2.
- √(a ÷ b) = √a ÷ √b, so √(9/16) = 3/4.
- √(a + b) is not √a + √b: √(9 + 16) = 5, not 7.
- √(x²) = |x|, the positive value.
- The square root of a negative number is imaginary: √−9 = 3i.
Square root calculator with solution: √72 step by step
The default input 72 shows how the square root calculator with solution works. Prime factors: 72 = 2 × 2 × 2 × 3 × 3. Pair them: (2 × 2)(3 × 3) × 2. Take one from each pair outside: 2 × 3 = 6. Answer: √72 = 6√2 ≈ 8.4853. Want an exact answer? The calculator shows the simplest radical form, 6√2, which is the square root with no decimals.
Square root calculator for fractions
For a square root calculator with fractions, take the root of the top and bottom separately: √(25/49) = 5/7. If the bottom is not a perfect square, rationalise it: √(2/3) = √6 ÷ 3 ≈ 0.8165. Enter the fraction as a decimal (2 ÷ 3 = 0.6667) to check.
Square root calculator with variables and equations
With variables, halve the exponents: √(x⁴y²) = x²|y| and √(50x³) = 5x√(2x) for x ≥ 0. To solve a square root equation such as √(x + 3) = 5, square both sides: x + 3 = 25, so x = 22, then check the answer in the original equation.
Square root calculator for quadratic equations
Square roots solve quadratic equations of the form x² = k: x = ±√k. x² = 50 gives x = ±7.071. For the full ax² + bx + c = 0 form, use the quadratic equation calculator, which takes the square root of the discriminant.
Square root inverse: calculator tip
The inverse of a square root is squaring: if √x = 9, then x = 81. Use the exponent calculator to square or raise to any power.
Square root without calculator: how to do it
A quick square root calculator trick for mental maths: find the two perfect squares around the number and interpolate. For √40: 36 < 40 < 49, so √40 is 6 plus (40 − 36) ÷ (2 × 6) = 6.33. The true value is 6.325. For exact digits, use the long division method above.
Square root key: calculator, TI-84 and iPhone
On most calculators the square root key or button is marked √. On basic calculators, type the number first and then press √. On a TI-84 calculator, press 2nd then x² to get √(, type the number and close the bracket. On an iPhone, open the Calculator app, rotate to landscape or switch to the scientific layout, and use the √x key. No square root calculator app is needed: this page works in any browser.
Square root calculator for flow and transmitter readings
In flow measurement with an orifice plate, flow is proportional to the square root of the differential pressure: Q = k × √ΔP. That is why a DP transmitter often applies square root extraction. At 25% of full-scale ΔP, flow is √0.25 = 50% of full flow; at 64% ΔP, flow is 80%. This square root calculation for a transmitter is why readings near zero flow are less precise.
Square root calculator curve for grades
Some teachers curve grades with a square root: new score = √(old score) × 10. A 64 becomes 80, a 81 becomes 90 and a 100 stays 100. The square root curve lifts low scores more than high ones.
Square root calculation in Java and Python
For a square root calculation in Java, use Math.sqrt(72), which returns 8.48528137423857. A square root calculator in Python uses math.sqrt(72), or math.isqrt(72) for the whole-number part (8).
Free square root worksheets: practice questions
- √144 = 12
- Simplify √50 → 5√2
- Simplify √98 → 7√2
- √(36/81) = 2/3
- Between which two whole numbers is √70? 8 and 9
Frequently asked questions
How do you calculate the square root of a number?
Find the number that multiplied by itself gives the original number. Use the prime factorization or long division method by hand, or enter the number in this square root calculator for the answer with steps.
What is the square root of 2?
The square root of 2 is about 1.41421356. It is irrational, so its decimal expansion never ends or repeats.
How do I simplify a square root?
Break the number into prime factors, take one factor out for every pair, and leave the unpaired factors inside. For example √72 = √(2² × 3² × 2) = 6√2.
Can you take the square root of a negative number?
Not as a real number. The result is imaginary: √(−16) = 4i. Odd roots such as cube roots of negative numbers are real, for example ∛(−8) = −2.
How do I find the cube root?
Choose Cube root in the calculator. By hand, group prime factors in threes: ∛216 = ∛(2³ × 3³) = 6.
What is the square root long division method?
It is a hand method that pairs the digits of the number, finds the largest square for the first pair, then repeatedly doubles the root so far and finds the next digit. It gives exact digits one at a time.
How do I find the square root on a calculator?
Type the number and press the √ key. On phones, rotate to landscape to see the scientific keys. This online calculator also shows the simplified form and steps.
Is this square root calculator free?
Yes. It is free, works online without an app, and also handles cube roots and nth roots.
How do I find the square root of a fraction?
Take the square root of the numerator and denominator separately. √(16/25) = 4/5.
Why does a flow transmitter use square root extraction?
Because flow through an orifice is proportional to the square root of the pressure drop. 25% of full ΔP means 50% of full flow.
What does the square root value calculator show?
It shows the decimal square root value to many places, the simplest radical form and the steps. For 50: 7.0711 and 5√2.
Can I use the square root calculator to simplify radicals?
Yes. The square root calculator simplify step pairs prime factors, so √200 becomes 10√2.
Is there a square root calculator with steps free to use?
Yes. This page is free with no sign-up, and every answer shows the prime factorization steps.
What does the square root calculator symbol mean?
The √ sign is the radical. The square root calculator symbol with a small number, such as ∛, means a cube or higher root.
Where is the square root key on a calculator?
Look for √. On many models it is a second function above the x² key.
Can the square root calculator solve an equation?
It finds roots of numbers. For a square root calculator equation such as √x = 7, square both sides: x = 49.
Last updated 2026-09-18