Calculate the square root, cube root, or any nth root of any number instantly. Shows decimal result, simplified radical form, step-by-step solution, squared value, and perfect square detection.
⚠️ Disclaimer: Results use IEEE 754 double-precision floating point and are shown to 10 decimal places. For exact symbolic results, simplified radical form is shown where possible.
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Sources & Methodology
✓Square root computed using JavaScript’s Math.sqrt() (IEEE 754 double precision). Simplified radical form computed by factoring out largest perfect square. Cube root and nth root use Math.pow(x, 1/n) with sign handling for odd roots of negatives.
U.S. government reference for mathematical functions including power functions, roots, and irrational number definitions. Pi and square roots of 2, 3, 5 are listed to full precision.
Methods used: Square root: sqrt(x) = x^(1/2) [Math.sqrt(x)]Cube root: cbrt(x) = x^(1/3), sign(x) * |x|^(1/3) for negativesNth root: x^(1/n), sign handling for odd n with negative xSimplified radical: find largest perfect square factor k, result = sqrt(k) * sqrt(x/k)
Precision: IEEE 754 double (15–17 significant decimal digits). Displayed to 10 decimal places.
Last reviewed: April 2026
Complete Guide to Square Roots, Cube Roots & Radicals
The square root is one of the most fundamental operations in mathematics. It is the inverse of squaring a number: if 5² = 25, then √25 = 5. Square roots appear throughout geometry (Pythagorean theorem), physics (distance and velocity formulas), statistics (standard deviation), finance (compound interest), and everyday calculations like finding the side length of a square from its area.
Key Square Root Facts
√25 = 5 (because 5 × 5 = 25)
√2 ≈ 1.41421356 (irrational — goes on forever)
√0 = 0 | √1 = 1 | √100 = 10
√50 = 5√2 ≈ 7.071 (simplified radical form)
(−5)² = 25, so −5 is also a square root of 25 (negative root)
Perfect Squares from 1 to 225
These are the integers whose square root is also an integer. Memorizing them speeds up mental math significantly:
1
1²
4
2²
9
3²
16
4²
25
5²
36
6²
49
7²
64
8²
81
9²
100
10²
121
11²
144
12²
169
13²
196
14²
225
15²
How to Simplify Square Roots (Simplified Radical Form)
A square root is in simplified radical form when the number inside has no perfect square factors other than 1. Process: (1) find the largest perfect square factor of the number, (2) rewrite as a product, (3) take the square root of the perfect square factor and move it outside the radical.
√50 = √(25 × 2) = √25 × √2 = 5√2
√72 = √(36 × 2) = √36 × √2 = 6√2
√98 = √(49 × 2) = √49 × √2 = 7√2
√75 = √(25 × 3) = √25 × √3 = 5√3
√200 = √(100 × 2) = 10√2 ≈ 14.142
Square Root of Common Numbers
Number
Square Root
Simplified Radical
Type
2
1.41421356…
√2
Irrational
3
1.73205080…
√3
Irrational
4
2.00000000
2
Perfect square
5
2.23606797…
√5
Irrational
8
2.82842712…
2√2
Simplified radical
12
3.46410161…
2√3
Simplified radical
25
5.00000000
5
Perfect square
50
7.07106781…
5√2
Simplified radical
Cube Roots — When Three Times is the Power
The cube root of x is the number that, when cubed, gives x. Written as ∛x or x^(1/3). Unlike square roots, cube roots exist for negative numbers: ∛(−27) = −3 because (−3)³ = −27. Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 (1³ through 10³). The cube root appears in volume-related calculations: if a cube has volume 64 cm³, its side length = ∛64 = 4 cm.
Nth Roots and Rational Exponents
The nth root of x equals x^(1/n). This connects radicals to exponent notation. The 4th root of 81 = 81^(0.25) = 3 (because 3&sup4; = 81). The 5th root of 32 = 32^(0.2) = 2 (because 2&sup5; = 32). For odd values of n (3, 5, 7…), negative numbers have real nth roots. For even n, negative numbers produce imaginary results. The general rule: (⊃n√x)⊃n; = x for any real x when n is odd, and for x ≥ 0 when n is even.
💡 Square Root in the Pythagorean Theorem: The most common real-world use of square roots. For a right triangle with legs a and b, hypotenuse c = √(a² + b²). A 3–4–5 right triangle: c = √(9+16) = √25 = 5. For a TV screen 40 inches wide and 22.5 inches tall: diagonal = √(1600 + 506.25) = √2106.25 = 45.9 inches (roughly 46″). This is how TV screen sizes are measured.
Square Root Properties and Rules
√(a × b) = √a × √b — product rule
√(a / b) = √a / √b — quotient rule
(√a)² = a — squaring cancels the root (for a ≥ 0)
√(a²) = |a| — result is always non-negative
√a + √b ≠ √(a+b) — common mistake! e.g. √9 + √16 = 3+4=7, NOT √25=5
Real-World Applications of Square Roots
Construction & carpentry: Finding diagonal measurements using the Pythagorean theorem
Physics: Speed from kinetic energy (v = √(2KE/m)), pendulum period
Statistics: Standard deviation is the square root of variance
Computer graphics: Distance between points, vector normalization
Engineering: RMS (Root Mean Square) values for AC electricity
Navigation: Straight-line distance between two coordinates
Frequently Asked Questions
The square root of x is a value that when multiplied by itself equals x. Written as √x or x^(1/2). Example: √25 = 5 because 5×5 = 25. Every positive number has two square roots: +5 and −5 (both satisfy the equation).
A perfect square is an integer with an integer square root. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. All are squares of whole numbers (1² through 12², etc.). Numbers like 2, 3, 5, 7 are NOT perfect squares — their square roots are irrational.
Simplified radical form extracts perfect square factors from under the root. Example: √50 = √(25×2) = 5√2. The number inside the radical (2) contains no perfect square factors other than 1. This is the standard simplified form required in algebra.
√2 ≈ 1.41421356. It is irrational (non-terminating, non-repeating decimal). It equals the length of the diagonal of a 1×1 square. It cannot be expressed as a fraction. Proved to be irrational by the ancient Greeks.
Not in real number arithmetic. √(−1) is undefined as a real number. In complex mathematics, √(−1) = i (imaginary unit). So √(−25) = 5i. This calculator shows when a result is imaginary.
∛x is a value that cubed equals x. ∛27 = 3 (3³=27). ∛(−8) = −2 ((−2)³=−8). Unlike square roots, cube roots exist for negative numbers since negative × negative × negative = negative.
c = √(a² + b²) finds the hypotenuse. Legs 6 and 8: c = √(36+64) = √100 = 10. It is the final step in solving for any unknown side of a right triangle.
√0 = 0. Because 0 × 0 = 0. Zero is technically a perfect square. It is the only number whose square root equals itself (along with 1).
For perfect squares, recognize them from memory. For others, use estimation: find the two perfect squares it falls between. √50 is between √49=7 and √64=8. Since 50 is much closer to 49, estimate ≈ 7.07. Newton’s method gives more precision: guess, divide, average, repeat.
They are NOT equal — this is the most common square root mistake. √(9+16) = √25 = 5. But √9 + √16 = 3 + 4 = 7. The product rule works (√(a×b) = √a × √b) but the addition rule does NOT.
Standard deviation = √(variance). Variance is the average of squared differences from the mean. Taking the square root brings the unit back to the original scale. This is why standard deviation is in the same unit as the data, while variance is in squared units.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100. These are 1² through 10². Extended to 225: add 121(11²), 144(12²), 169(13²), 196(14²), 225(15²).