Square Root Calculator (√) | Simple & Precise Sqrt Tool
Calculate square roots, cube roots, or custom degree radicals. Simplify surds with prime factor pairings and visualize manual division step-by-step.
🔬 Select Radical Mode
🔄 Bidirectional Calculation Console
Enter a value in either field. The opposite value will synchronize instantly.
💡 Standard Mathematical Constants
📊 Perfect Powers & Neighbor Bounds
The base 144 is a perfect integer power of degree 2 (i.e. 122 = 144).
🌳 Step-by-Step Prime Factor Surd Simplification
To simplify radicals, we decompose the number into prime factors, group them according to degree, and pull perfect powers out of the radical.
Prime Factorization
Decompose 144 into its prime base components:
Factor Exponent Grouping (Degree 2)
Group prime factors into perfect powers matching our radical degree:
Extract Perfect Exponents
Extract powers and simplify terms outside and inside the radical:
✏️ Long Division Square Root Board
Chalk Board StepsManual digit-by-digit extraction. Digits are paired from the decimal point. We calculate double quotients and subtract.
📖 Reference Table: Square & Cube Roots (1 to 100)
Quickly reference integers from 1 to 100, including exact surds and perfect square markers.
| Number (X) | Square Root (√x) | Cube Root (³√x) | Simplified Surd (Degree 2) | Perfect Square? |
|---|---|---|---|---|
| 1 | 1.0000 | 1.0000 | √1 | Yes (Perfect) |
| 2 | 1.4142 | 1.2599 | √2 | No |
| 3 | 1.7321 | 1.4422 | √3 | No |
| 4 | 2.0000 | 1.5874 | 2 | Yes (Perfect) |
| 5 | 2.2361 | 1.7100 | √5 | No |
| 6 | 2.4495 | 1.8171 | √6 | No |
| 7 | 2.6458 | 1.9129 | √7 | No |
| 8 | 2.8284 | 2.0000 | 2√2 | No |
| 9 | 3.0000 | 2.0801 | 3 | Yes (Perfect) |
| 10 | 3.1623 | 2.1544 | √10 | No |
| 11 | 3.3166 | 2.2240 | √11 | No |
| 12 | 3.4641 | 2.2894 | 2√3 | No |
| 13 | 3.6056 | 2.3513 | √13 | No |
| 14 | 3.7417 | 2.4101 | √14 | No |
| 15 | 3.8730 | 2.4662 | √15 | No |
| 16 | 4.0000 | 2.5198 | 4 | Yes (Perfect) |
| 17 | 4.1231 | 2.5713 | √17 | No |
| 18 | 4.2426 | 2.6207 | 3√2 | No |
| 19 | 4.3589 | 2.6684 | √19 | No |
| 20 | 4.4721 | 2.7144 | 2√5 | No |
| 21 | 4.5826 | 2.7589 | √21 | No |
| 22 | 4.6904 | 2.8020 | √22 | No |
| 23 | 4.7958 | 2.8439 | √23 | No |
| 24 | 4.8990 | 2.8845 | 2√6 | No |
| 25 | 5.0000 | 2.9240 | 5 | Yes (Perfect) |
| 26 | 5.0990 | 2.9625 | √26 | No |
| 27 | 5.1962 | 3.0000 | 3√3 | No |
| 28 | 5.2915 | 3.0366 | 2√7 | No |
| 29 | 5.3852 | 3.0723 | √29 | No |
| 30 | 5.4772 | 3.1072 | √30 | No |
| 31 | 5.5678 | 3.1414 | √31 | No |
| 32 | 5.6569 | 3.1748 | 4√2 | No |
| 33 | 5.7446 | 3.2075 | √33 | No |
| 34 | 5.8310 | 3.2396 | √34 | No |
| 35 | 5.9161 | 3.2711 | √35 | No |
| 36 | 6.0000 | 3.3019 | 6 | Yes (Perfect) |
| 37 | 6.0828 | 3.3322 | √37 | No |
| 38 | 6.1644 | 3.3620 | √38 | No |
| 39 | 6.2450 | 3.3912 | √39 | No |
| 40 | 6.3246 | 3.4200 | 2√10 | No |
| 41 | 6.4031 | 3.4482 | √41 | No |
| 42 | 6.4807 | 3.4760 | √42 | No |
| 43 | 6.5574 | 3.5034 | √43 | No |
| 44 | 6.6332 | 3.5303 | 2√11 | No |
| 45 | 6.7082 | 3.5569 | 3√5 | No |
| 46 | 6.7823 | 3.5830 | √46 | No |
| 47 | 6.8557 | 3.6088 | √47 | No |
| 48 | 6.9282 | 3.6342 | 4√3 | No |
| 49 | 7.0000 | 3.6593 | 7 | Yes (Perfect) |
| 50 | 7.0711 | 3.6840 | 5√2 | No |
| 51 | 7.1414 | 3.7084 | √51 | No |
| 52 | 7.2111 | 3.7325 | 2√13 | No |
| 53 | 7.2801 | 3.7563 | √53 | No |
| 54 | 7.3485 | 3.7798 | 3√6 | No |
| 55 | 7.4162 | 3.8030 | √55 | No |
| 56 | 7.4833 | 3.8259 | 2√14 | No |
| 57 | 7.5498 | 3.8485 | √57 | No |
| 58 | 7.6158 | 3.8709 | √58 | No |
| 59 | 7.6811 | 3.8930 | √59 | No |
| 60 | 7.7460 | 3.9149 | 2√15 | No |
| 61 | 7.8102 | 3.9365 | √61 | No |
| 62 | 7.8740 | 3.9579 | √62 | No |
| 63 | 7.9373 | 3.9791 | 3√7 | No |
| 64 | 8.0000 | 4.0000 | 8 | Yes (Perfect) |
| 65 | 8.0623 | 4.0207 | √65 | No |
| 66 | 8.1240 | 4.0412 | √66 | No |
| 67 | 8.1854 | 4.0615 | √67 | No |
| 68 | 8.2462 | 4.0817 | 2√17 | No |
| 69 | 8.3066 | 4.1016 | √69 | No |
| 70 | 8.3666 | 4.1213 | √70 | No |
| 71 | 8.4261 | 4.1408 | √71 | No |
| 72 | 8.4853 | 4.1602 | 6√2 | No |
| 73 | 8.5440 | 4.1793 | √73 | No |
| 74 | 8.6023 | 4.1983 | √74 | No |
| 75 | 8.6603 | 4.2172 | 5√3 | No |
| 76 | 8.7178 | 4.2358 | 2√19 | No |
| 77 | 8.7750 | 4.2543 | √77 | No |
| 78 | 8.8318 | 4.2727 | √78 | No |
| 79 | 8.8882 | 4.2908 | √79 | No |
| 80 | 8.9443 | 4.3089 | 4√5 | No |
| 81 | 9.0000 | 4.3267 | 9 | Yes (Perfect) |
| 82 | 9.0554 | 4.3445 | √82 | No |
| 83 | 9.1104 | 4.3621 | √83 | No |
| 84 | 9.1652 | 4.3795 | 2√21 | No |
| 85 | 9.2195 | 4.3968 | √85 | No |
| 86 | 9.2736 | 4.4140 | √86 | No |
| 87 | 9.3274 | 4.4310 | √87 | No |
| 88 | 9.3808 | 4.4480 | 2√22 | No |
| 89 | 9.4340 | 4.4647 | √89 | No |
| 90 | 9.4868 | 4.4814 | 3√10 | No |
| 91 | 9.5394 | 4.4979 | √91 | No |
| 92 | 9.5917 | 4.5144 | 2√23 | No |
| 93 | 9.6437 | 4.5307 | √93 | No |
| 94 | 9.6954 | 4.5468 | √94 | No |
| 95 | 9.7468 | 4.5629 | √95 | No |
| 96 | 9.7980 | 4.5789 | 4√6 | No |
| 97 | 9.8489 | 4.5947 | √97 | No |
| 98 | 9.8995 | 4.6104 | 7√2 | No |
| 99 | 9.9499 | 4.6261 | 3√11 | No |
| 100 | 10.0000 | 4.6416 | 10 | Yes (Perfect) |
Overview & Capabilities
Our **Square Root Calculator** is a precision tool designed to find the square root (√) of any positive number instantly. Whether you are solving academic math problems, engineering equations, or curiosity-driven calculations, our tool provides high-accuracy results including perfect square markers and decimal expansions.
How to Use
Key Features
Common Use Cases
Tips & Best Practices
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Frequently Asked Questions
Q What is a square root?
A square root of a number x is a number y such that y² = x. For example, the square root of 9 is 3 because 3 × 3 = 9.
Q Can a negative number have a square root?
In the real number system, negative numbers do not have square roots because the square of any real number is non-negative. In complex mathematics, they are represented using the imaginary unit "i".
Q Is √2 a rational number?
No, √2 is an irrational number, approximately 1.414. Its decimal representation goes on forever without repeating.