Square Root Calculator (√) & Radical Surd Studio
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
Calculate principal roots and reduce radical surds into simplified radical form with our Square Root Studio! Extract largest perfect square factors to simplify expressions (e.g. \(\sqrt{108} = 6\sqrt{3}\)), evaluate arbitrary nth roots (\(\sqrt[n]{x}\)), and compute imaginary complex solutions (\(\sqrt{-x} = i\sqrt{x}\)).
How to Use
Key Features
Common Use Cases
Tips & Best Practices
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Frequently Asked Questions
Q How do you simplify an unreduced radical like √98?
1. Identify the largest perfect square factor of 98 (which is 49). 2. Express 98 as 49 × 2. 3. Apply the radical product rule: √49 × √2 = 7√2.
Q What distinguishes a rational square root from an irrational root?
The square root of a perfect square (like √36 = 6) is a rational number. The square root of any non-perfect square integer (like √2 or √5) is irrational, producing an infinite non-repeating decimal.
Q Why does √x denote only the principal positive square root?
By mathematical convention, the radical symbol √ denotes the non-negative principal root so that the square root function f(x) = √x remains a well-defined mathematical function with a unique single output.