√ Enter Your Number
Choose root type and enter any number
Quick — Click a perfect square:
√4
= 2
√9
= 3
√16
= 4
√25
= 5
√36
= 6
√49
= 7
√64
= 8
√81
= 9
√100
= 10
√121
= 11
√144
= 12
√169
= 13
📊 Your Result
✅ Perfect Square!
√144 =
12
Square Root
Also expressed as
12.0000 (exact)
✅ Perfect Square — 12 × 12 = 144
🌟 Perfect Square!
144 is a perfect square! The exact answer is 12 with no rounding needed.
📋 Step-by-Step Working
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Square Root Calculator — Complete Guide
The square root is one of the most important operations in mathematics. It appears in geometry, physics, statistics, engineering, computer science and everyday calculations. Understanding square roots — how to calculate them, what they mean and where they apply — is a fundamental mathematical skill that builds the foundation for algebra, calculus and beyond.
What Is a Square Root?
The square root of a number n is the value x such that x × x = n. For example the square root of 25 is 5 because 5 × 5 = 25. The square root symbol √ is called a radical sign. Every positive number has two square roots: a positive root (called the principal root) and a negative root. The number under the radical sign is called the radicand.
Square Root Definition:
√n = x where x × x = n
Examples:
√4 = 2 because 2 × 2 = 4
√9 = 3 because 3 × 3 = 9
√16 = 4 because 4 × 4 = 16
√25 = 5 because 5 × 5 = 25
√2 = 1.4142... (irrational - never ends)
√3 = 1.7320... (irrational - never ends)
Negative roots also exist:
√25 = +5 OR -5 (both 5×5 and (-5)×(-5) = 25)
By convention √ symbol means the positive root
Perfect Squares 1 to 30 — Complete Reference Table
A perfect square is an integer whose square root is also an integer. Memorising perfect squares up to 30 significantly speeds up mental arithmetic, exam performance and everyday calculations.
| Number | Square Root | Number | Square Root | Number | Square Root |
|---|---|---|---|---|---|
| 1 | 1 | 121 | 11 | 441 | 21 |
| 4 | 2 | 144 | 12 | 484 | 22 |
| 9 | 3 | 169 | 13 | 529 | 23 |
| 16 | 4 | 196 | 14 | 576 | 24 |
| 25 | 5 | 225 | 15 | 625 | 25 |
| 36 | 6 | 256 | 16 | 676 | 26 |
| 49 | 7 | 289 | 17 | 729 | 27 |
| 64 | 8 | 324 | 18 | 784 | 28 |
| 81 | 9 | 361 | 19 | 841 | 29 |
| 100 | 10 | 400 | 20 | 900 | 30 |
Cube Roots and Nth Roots Explained
Beyond square roots mathematics uses higher-order roots. A cube root finds the number that multiplied by itself three times gives the radicand. An nth root generalises this to any power. Our calculator handles all these cases with the same instant accuracy.
| Root Type | Symbol | Example | Common Use |
|---|---|---|---|
| Square Root (2nd) | √ | √25 = 5 | Geometry, Pythagorean theorem |
| Cube Root (3rd) | ∛ | ∛27 = 3 | Volume of cubes, physics |
| 4th Root | ⁴√ | ⁴√81 = 3 | Statistics, compound interest |
| Nth Root | n√ | 5√32 = 2 | Advanced mathematics, engineering |
How to Simplify Square Roots
Simplifying a square root means expressing it in its simplest radical form. The method involves finding perfect square factors of the radicand and pulling them out of the radical sign.
Simplifying Square Roots:
Step 1: Find the largest perfect square factor
Step 2: Split the radical into two parts
Step 3: Take the square root of the perfect square factor
Example: Simplify √72
Factors of 72 = 36 × 2
√72 = √(36 × 2) = √36 × √2 = 6√2
Example: Simplify √200
Factors: 200 = 100 × 2
√200 = √100 × √2 = 10√2
Example: Simplify √48
Factors: 48 = 16 × 3
√48 = √16 × √3 = 4√3
Real-World Applications of Square Roots
- Pythagorean theorem: Finding the hypotenuse or any side of a right triangle requires square roots. Distance = √(a² + b²). Used in construction, navigation, architecture and everyday distance calculations.
- Statistics and data analysis: Standard deviation — the most important measure of data spread — involves taking the square root of the variance. Every statistics course and data science project uses this constantly.
- Physics and engineering: Velocity from kinetic energy uses √(2E/m). Frequency calculations, electrical impedance, wave equations and structural load calculations all involve square roots. Use our Scientific Calculator for complex physics calculations!
- Finance: The compound annual growth rate (CAGR) formula uses nth roots. Volatility in finance is measured as the square root of variance. Our Compound Interest Calculator uses these formulas internally!
- Computer graphics and games: Calculating the distance between two points on screen requires the Pythagorean theorem and square roots. Every 3D game engine calls square root functions thousands of times per second.
- Geometry: Finding the side of a square from its area requires a square root. A square with area 64 has sides of length √64 = 8. Use our Percentage Calculator for related geometric calculations!
Common Square Roots Students Need to Know
For exams and everyday mental maths these are the most important square root values to memorise. They come up repeatedly in algebra, geometry and standardised tests like SAT, ACT, GCSE and A-Level Mathematics.
√1 = 1 √2 = 1.414 √3 = 1.732
√4 = 2 √5 = 2.236 √6 = 2.449
√7 = 2.646 √8 = 2.828 √9 = 3
√10 = 3.162 √50 = 7.071 √100 = 10
√144 = 12 √169 = 13 √196 = 14
√225 = 15 √256 = 16 √289 = 17
√324 = 18 √361 = 19 √400 = 20
📚 Note: Square root calculations shown are correct to 10 decimal places. For very large numbers or numbers requiring more than 10 decimal places of precision use dedicated scientific computing software. Irrational square roots like √2 and √3 are infinite non-repeating decimals — our calculator shows the first 6 significant figures.
Frequently Asked Questions
What is the square root of a number? +
The square root of a number is a value that when multiplied by itself gives the original number. For example the square root of 25 is 5 because 5 × 5 = 25. The symbol √ is called a radical sign. Every positive number has two square roots: a positive root (principal root) and a negative root. The square root of 25 is both +5 and -5 since both squared give 25. By convention the radical symbol refers to the positive root.
How do you calculate a square root by hand? +
For perfect squares simply identify which whole number squared gives the result. For non-perfect squares use estimation: find the two perfect squares the number falls between. For √20 — it falls between 16 and 25, so between 4 and 5. Refine: try 4.4 (4.4²=19.36), try 4.5 (4.5²=20.25). So √20 is between 4.4 and 4.5. The exact answer is 4.4721. The long division method can find any number of decimal places but our calculator does this instantly.
What is the difference between a perfect square and an irrational square root? +
A perfect square is a number whose square root is a whole integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on. Their roots are exactly 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. An irrational square root is a decimal that continues forever without repeating — like √2 = 1.41421356... or √3 = 1.73205080... These cannot be expressed as exact fractions and must be approximated for practical use.
What is a cube root? +
A cube root is the number that when multiplied by itself three times gives the original number. The cube root of 27 is 3 because 3 × 3 × 3 = 27. The cube root of 8 is 2 because 2 × 2 × 2 = 8. Unlike square roots cube roots can be negative: the cube root of -27 is -3 because (-3) × (-3) × (-3) = -27. Our calculator handles cube roots using the Nth root with n=3.
Can you take the square root of a negative number? +
In the real number system there is no square root of a negative number because no real number multiplied by itself gives a negative result. However in mathematics imaginary numbers handle this: the square root of -1 is defined as i (imaginary unit). The square root of -9 = 3i. Imaginary numbers are essential in electrical engineering, quantum physics and signal processing but are beyond standard school mathematics calculators.
What is an nth root? +
An nth root is the generalisation of square and cube roots. The nth root of x is the value that when raised to the power n gives x. The 4th root of 81 is 3 because 3&sup4; = 81. The 5th root of 32 is 2 because 2&sup5; = 32. In financial mathematics the nth root is used to calculate compound annual growth rate (CAGR) from start and end values over n years.
Where are square roots used in real life? +
Square roots appear in many practical situations: the Pythagorean theorem for calculating distances (distance = √(a²+b²)), finding the side length of a square from its area, standard deviation in statistics, physics formulas for velocity and energy, computer graphics for point-to-point distances, finance for volatility and CAGR calculations, and engineering for structural and electrical calculations.
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