Square Root Calculator
How Square Roots Work
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 144 is 12, because 12 times 12 equals 144. This calculator also supports cube roots (a number multiplied by itself three times) and higher-order roots. Need to continue this calculation? Try the Exponent Calculator or the Scientific Calculator.
Perfect Squares
A perfect square is a number whose square root is a whole number. Common perfect squares include 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225. Most numbers are not perfect squares and have irrational square roots (decimals that never repeat or terminate).
How can you apply these results practically?
The Square Root Calculator is built for people who want a fast answer without losing context. It keeps the calculation simple, shows the result clearly, and helps you understand what the number means before you use it in a real decision.
This calculator is designed to make a specific everyday calculation faster and clearer. It gives a structured result so you can compare options, check assumptions, or plan the next step with less manual work.
What is a square root?
The square root of a number n is the value that, when multiplied by itself, gives n. Written as √n. For example, √25 = 5 because 5 × 5 = 25. Square roots appear in geometry (calculating side lengths from areas), physics (distance formulas), finance (standard deviation), and algebra throughout mathematics education.
Every positive number has two square roots: a positive root (+√n) and a negative root (–√n). By convention, √n refers to the principal (positive) square root. Negative numbers do not have real square roots — √(–1) is defined as the imaginary unit i in complex mathematics.
Perfect squares — instant recognition
Perfect squares are integers whose square roots are also integers. Recognising these instantly speeds up mental arithmetic:
- √1 = 1 | √4 = 2 | √9 = 3 | √16 = 4 | √25 = 5
- √36 = 6 | √49 = 7 | √64 = 8 | √81 = 9 | √100 = 10
- √121 = 11 | √144 = 12 | √169 = 13 | √196 = 14 | √225 = 15
- √256 = 16 | √289 = 17 | √324 = 18 | √361 = 19 | √400 = 20
How to calculate square roots manually
Method 1 — Estimation and refinement: For √50, note that 7² = 49 and 8² = 64. So √50 is between 7 and 8, closer to 7. Try 7.1²= 50.41 — too high. Try 7.07² = 49.98 — very close. √50 ≈ 7.071
Method 2 — Prime factorisation (for perfect squares): √144 = √(4 × 36) = √4 × √36 = 2 × 6 = 12. Or: 144 = 2⁴ × 3². √(2⁴ × 3²) = 2² × 3 = 4 × 3 = 12.
Method 3 — Long division method: A systematic algorithm for computing square roots to any decimal precision, used in manual calculation before calculators.
Cube roots and higher roots
The cube root (∛n) is the value that when cubed gives n. ∛27 = 3 because 3³ = 27. ∛64 = 4. ∛1000 = 10.
The nth root is written as ⁿ√n or n^(1/n) in exponential notation:
- Square root: n^(1/2) → √25 = 25^0.5 = 5
- Cube root: n^(1/3) → ∛27 = 27^(1/3) = 3
- Fourth root: n^(1/4) → ⁴√16 = 16^0.25 = 2
Square roots in practical applications
Pythagoras theorem: In a right triangle, c² = a² + b². The hypotenuse c = √(a² + b²). For a 3–4–5 triangle: c = √(9 + 16) = √25 = 5. This is used in construction to verify right angles and calculate diagonal distances.
Distance formula: Distance between points (x₁,y₁) and (x₂,y₂) = √[(x₂–x₁)² + (y₂–y₁)²]. This is the Pythagorean theorem applied to coordinate geometry.
Standard deviation: The standard deviation is the square root of the variance. If variance = 16, standard deviation = √16 = 4. Square roots convert the squared units back to original units.
Compound interest: To find the interest rate that doubles money in n years: Rate = (2)^(1/n) – 1. To double in 6 years: Rate = 2^(1/6) – 1 = 1.1225 – 1 = 12.25% CAGR. This uses fractional exponents equivalent to roots.
Simplifying square root expressions
√(a × b) = √a × √b — the product rule for square roots.
- √72 = √(36 × 2) = √36 × √2 = 6√2 ≈ 6 × 1.414 = 8.485
- √200 = √(100 × 2) = 10√2 ≈ 14.142
- √(9/16) = √9 / √16 = 3/4 = 0.75
Perfect squares: 1 to 400
A perfect square is an integer whose square root is also an integer. The first 20 perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400. Key pattern: the difference between consecutive perfect squares increases by 2 each time (1, 3, 5, 7, 9...). This pattern is the basis of the sieve of Eratosthenes for finding prime numbers.
Irrational square roots and decimal approximation
Non-perfect squares produce irrational decimal roots — they never terminate or repeat. √2 = 1.41421356... √3 = 1.73205081... √5 = 2.23606798... These cannot be expressed as fractions. For approximation without a calculator: √(n) ≈ √(nearest perfect square) + (n − perfect square) ÷ (2 × √nearest perfect square). Estimate √50: nearest square is 49. √50 ≈ 7 + 1/14 ≈ 7.07 (actual: 7.071).
Cube roots and nth roots
Cube root (∛x) finds the number that, multiplied by itself three times, gives x. ∛27 = 3 because 3³ = 27. ∛125 = 5. ∛8 = 2. Fractional exponents express nth roots: ∛x = x^(1/3). Square root = x^(1/2). Fourth root = x^(1/4). Negative cube roots exist (unlike square roots): ∛(−27) = −3 because (−3)³ = −27. Cube roots are used in volume calculations (edge length from volume).
Frequently asked questions
What is the square root of 2? √2 ≈ 1.41421356… It is an irrational number — its decimal representation never terminates or repeats. It appears in the diagonal of a unit square (by Pythagoras: √(1² + 1²) = √2) and in many geometric and trigonometric contexts.
Can a square root be negative? The principal (standard) square root of a positive number is always positive. However, equations like x² = 25 have two solutions: x = +5 and x = –5. In complex numbers, √(–1) = i (the imaginary unit).
What is the square root of 0? √0 = 0. Zero is its own square root since 0 × 0 = 0.
Perfect squares reference table (1–400)
| Number | Square Root | Number | Square Root | Number | Square Root |
|---|---|---|---|---|---|
| 1 | 1 | 49 | 7 | 169 | 13 |
| 4 | 2 | 64 | 8 | 196 | 14 |
| 9 | 3 | 81 | 9 | 225 | 15 |
| 16 | 4 | 100 | 10 | 256 | 16 |
| 25 | 5 | 121 | 11 | 289 | 17 |
| 36 | 6 | 144 | 12 | 324 | 18 |
| 361 | 19 | ||||
| 400 | 20 |
Approximations for common irrational square roots
| Number | Square Root | Approx. | Application |
|---|---|---|---|
| 2 | √2 | 1.41421 | Diagonal of unit square; f-stops in photography |
| 3 | √3 | 1.73205 | Height of equilateral triangle; electronics (3-phase) |
| 5 | √5 | 2.23607 | Golden ratio = (1+√5)/2 ≈ 1.618 |
| 6 | √6 | 2.44949 | |
| 7 | √7 | 2.64575 | |
| 10 | √10 | 3.16228 | RMS of equal-probability binary values |
Sources & References
Sources: Hardy, G.H. A Course of Pure Mathematics (10th ed), Cambridge; NIST e-Handbook of Statistical Methods; Pythagoras theorem — Euclid, Elements, Book I.
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