Calculate square roots, simplify radicals using prime factorization, check perfect squares, compute nth roots, and perform radical operations — all with full step-by-step solutions.
The square root of a number x is the value that, when multiplied by itself, equals x. Written √x, it answers the question "what number squared gives x?" For example, √16 = 4 because 4 × 4 = 16. Every positive number has two square roots — a positive one (the principal root, what the √ symbol means by convention) and a negative one. √16 = 4, but −4 is also a root since (−4)² = 16.
Square roots are a special case of exponentiation: √x = x½. This connection makes many algebraic manipulations cleaner. The inverse of squaring is taking a square root; the inverse of cubing is taking a cube root; the inverse of raising to the nth power is taking the nth root. These are all unified under the notation x1/n.
Computes the principal square root of any non-negative number to your chosen precision. Also shows the simplified radical form (a√b) and confirms whether the input is a perfect square. Negative inputs are rejected — square roots of negative numbers are imaginary (involving i = √−1) and outside the scope of real-number calculation.
Computes any root: cube root (n=3), fourth root (n=4), or any integer n. For odd roots, negative inputs are allowed — ∛−8 = −2 because (−2)³ = −8. For even roots, negative inputs have no real solution. The calculator uses tolerance-based perfect root detection (checking whether rounding the result back to an integer and raising to the nth power recovers the original) to avoid floating-point false negatives.
A perfect square is a positive integer whose square root is also an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100... The checker also shows the two nearest perfect squares (floor and ceiling) so you can see how far the input is from being perfect, and shows the prime factorization — a perfect square will always have every prime factor appearing an even number of times in its factorization.
Simplifying √n means writing it as a√b where b has no perfect square factors — called simplest radical form. The algorithm uses prime factorization: factor n completely, then extract pairs of identical prime factors (each pair comes out of the radical as one factor), while unpaired primes remain inside. √72: factor 72 = 2³ × 3². Pairs: one pair of 2s, one pair of 3s. Outside: 2 × 3 = 6. Inside: one unpaired 2. Result: 6√2.
Multiplication: √a × √b = √(a×b). After computing the product under the radical, the result is fully simplified. √8 × √18 = √144 = 12.
Division: √a ÷ √b = √(a/b) when a/b is an integer. Otherwise the result is computed as a decimal with the exact rational under the radical shown. √50 ÷ √2 = √25 = 5.
Addition/Subtraction: Both radicals are first simplified to a√b form. If they share the same radicand (like radicals), the coefficients are combined: 2√3 + 5√3 = 7√3. Unlike radicals (√2 + √3) cannot be simplified — only decimal approximations combine.
| n | n² | √(n²) | n | n² | √(n²) |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 11 | 121 | 11 |
| 2 | 4 | 2 | 12 | 144 | 12 |
| 3 | 9 | 3 | 13 | 169 | 13 |
| 4 | 16 | 4 | 14 | 196 | 14 |
| 5 | 25 | 5 | 15 | 225 | 15 |
| 6 | 36 | 6 | 16 | 256 | 16 |
| 7 | 49 | 7 | 17 | 289 | 17 |
| 8 | 64 | 8 | 18 | 324 | 18 |
| 9 | 81 | 9 | 19 | 361 | 19 |
| 10 | 100 | 10 | 20 | 400 | 20 |
| Original | Simplified | Decimal | Key factor |
|---|---|---|---|
| √2 | √2 | 1.41421… | Prime — already simplest |
| √3 | √3 | 1.73205… | Prime — already simplest |
| √8 | 2√2 | 2.82842… | 8 = 4 × 2 |
| √12 | 2√3 | 3.46410… | 12 = 4 × 3 |
| √18 | 3√2 | 4.24264… | 18 = 9 × 2 |
| √20 | 2√5 | 4.47213… | 20 = 4 × 5 |
| √27 | 3√3 | 5.19615… | 27 = 9 × 3 |
| √32 | 4√2 | 5.65685… | 32 = 16 × 2 |
| √45 | 3√5 | 6.70820… | 45 = 9 × 5 |
| √48 | 4√3 | 6.92820… | 48 = 16 × 3 |
| √50 | 5√2 | 7.07106… | 50 = 25 × 2 |
| √72 | 6√2 | 8.48528… | 72 = 36 × 2 |
| √75 | 5√3 | 8.66025… | 75 = 25 × 3 |
| √98 | 7√2 | 9.89949… | 98 = 49 × 2 |
Long before calculators, mathematicians used an iterative algorithm to compute square roots by hand. Starting with any initial guess g, repeatedly replace g with the average of g and n/g. This converges quadratically — the number of correct digits roughly doubles with each iteration. To find √50: guess g = 7. Iteration 1: (7 + 50/7)/2 = (7 + 7.143)/2 = 7.071. Iteration 2: (7.071 + 50/7.071)/2 = 7.07106… — already 6 significant figures after just two steps.
Guess g₀ = 7
g₁ = (7 + 50/7) / 2 = 7.07143…
g₂ = (7.07143 + 50/7.07143) / 2 = 7.07107…
g₃ = 7.07106781… ← correct to 9 places
Square roots appear constantly in distance, area, and physics problems. The Pythagorean theorem uses √(a² + b²) to compute the hypotenuse. The distance formula in 2D space is √((x&sub2;−x&sub1;)² + (y&sub2;−y&sub1;)²). In physics, the RMS speed of gas molecules is proportional to √T (square root of absolute temperature). The period of a pendulum is proportional to √L (square root of length). Standard deviation is the square root of variance. Understanding how to compute and simplify square roots precisely matters in all of these contexts.