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.
Square roots appear in practical applications far more often than most people realize. In finance, standard deviation, the primary measure of investment risk, is the square root of variance. The Sharpe ratio, value at risk, and Black-Scholes options pricing all involve square roots. Annualizing monthly volatility multiplies by √12. In construction, the Pythagorean theorem (a² + b² = c²) requires a square root to find the hypotenuse: c = √(a² + b²). Every rafter length, diagonal brace, and stair calculation involves this. The classic 3-4-5 check for square corners: √(9 + 16) = √25 = 5, carpenters use this constantly. In everyday life, area scales as the square of linear dimensions. A room twice as wide AND twice as long has four times the area. Square roots reverse this relationship, which is why a 65-inch TV diagonal with a 16:9 aspect ratio has a width of √(65² × (16/√(16²+9²))²) ≈ 56.7 inches.
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.
Handles square roots, cube roots, and all radical expressions with visual display.
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