Square Root Calculator
Type a number to get its square root, check whether it is a perfect square, and see it in simplest surd form.
- Nothing you type is stored
- Free
- No signup
Root
8.4852813742
- Written as
- √72
- Simplest surd form
- 6√2
- Check
- 8.4852814² ≈ 72.00000044
Show working
- 72 = 2³ × 3²
- √72 = √(6² × 2) = 6√2
- = 6 × 1.414213562 = 8.485281374
How to use
- Keep Square √ selected (or choose cube or nth root).
- Type the number.
- Read the root, the simplest surd form and the prime-factor steps.
Good to know
The square root of x is the number that gives x when multiplied by itself: √144 = 12 because 12 × 12 = 144. Every positive number has two square roots (+12 and −12); the √ sign means the positive one. Negative numbers have no real square root.
Simplest surd form comes from prime factors: 72 = 2³ × 3², so √72 = √(6² × 2) = 6√2. Each pair of equal prime factors comes out of the root as one factor. This is how NCERT Class 8 (squares and square roots) and Class 9 (real numbers) present it.
Perfect squares up to 30² = 900 are worth learning for exams. The decimal value is shown to 10 places; for estimating by hand, note that √50 lies between 7 (√49) and 8 (√64), closer to 7.
Questions
What is the square root of 2?
About 1.4142135624. It is irrational, so the decimal never ends or repeats.
How do I simplify √48?
48 = 16 × 3 = 4² × 3, so √48 = 4√3.
How do I know if a number is a perfect square?
In its prime factorisation every exponent is even. 144 = 2⁴ × 3², so it is a perfect square. The calculator marks perfect squares.
What is the square root of a negative number?
It is not a real number. In higher maths √−1 is written i (an imaginary number), which this calculator does not cover.
How do I find a square root by the long division method?
Pair the digits from the decimal point, find the largest square under the first pair, then bring down pairs and double the root so far at each step. The answer here lets you check your working.