CALTRX

Square Root Calculator

Calculate the square root of any positive number with full decimal precision. Also checks whether the result is a perfect square.

Enter your values above to see the results.

Tips & Notes

  • ✓Always simplify using the LARGEST perfect square factor. √200 = √(100×2) = 10√2, not √(4×50) = 2√50.
  • ✓Estimate √n by finding the two perfect squares it falls between. √50: 7²=49, 8²=64 → answer is between 7 and 8, close to 7.
  • ✓x^(1/2) = √x and x^(1/3) = ∛x. So 64^(1/2) = 8 and 64^(1/3) = 4.
  • ✓Negative numbers have no real square root. √(−9) = 3i (imaginary). In physics a negative under √ means an error in setup.
  • ✓Standard calculators overflow at 171!. For large roots use scientific notation input.

Common Mistakes

  • ✗Adding square roots like numbers: √4 + √9 = 2+3 = 5, NOT √13. Square roots of different numbers cannot be added directly.
  • ✗Thinking √(a²+b²) = a+b. Pythagorean: √(3²+4²) = √25 = 5, NOT 3+4 = 7.
  • ✗Confusing squaring with multiplying by 2. √49 = 7 because 7² = 49, not 7×2 = 14.
  • ✗Not fully simplifying: √48 = 4√3, not 2√12. Check — can the number inside √ be simplified further?
  • ✗Taking √(a+b) = √a + √b. This is wrong. Only √(a×b) = √a × √b holds.

Square Root Calculator Overview

The square root of x is the non-negative number y such that y² = x. Square roots are the inverse of squaring — they answer the question "what number multiplied by itself gives this result?" They appear throughout geometry (Pythagorean theorem), statistics (standard deviation), physics (velocity and energy formulas), and engineering (diagonal calculations, signal processing). This calculator computes exact integer square roots, simplified radical forms, decimal approximations, cube roots, and nth roots.

The definition and principal root:

√x = y where y ≥ 0 and y² = x (the √ symbol always means the positive root)
EX: √81 = 9 (since 9² = 81) | √144 = 12 | √2 ≈ 1.41421356... | √(1/4) = 1/2
Simplifying square roots — extract the largest perfect square factor:
EX: √72 → factors: 4×18, 9×8, 36×2 → largest perfect square is 36 → √72 = √36 × √2 = 6√2 ≈ 8.485
EX: √200 → 100×2 → √200 = 10√2 ≈ 14.142 | EX: √108 → 36×3 → √108 = 6√3 ≈ 10.392
Always use the largest perfect square factor — using a smaller one requires additional simplification steps. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Product and quotient rules: - √(a×b) = √a × √b (valid for non-negative a, b) - √(a/b) = √a / √b (valid for a≥0, b>0)
EX: √(4×9) = √4 × √9 = 2×3 = 6 ✓ | √(9/16) = 3/4 = 0.75
Sign rules for roots: Even roots (√, ⁴√): input must be ≥ 0 in real numbers. √(−4) is undefined (no real number squared gives −4). - Odd roots (∛, ⁵√): negative input allowed. ∛(−8) = −2 because (−2)³ = −8. Estimating square roots mentally: find the two perfect squares bracketing your number. √50: 7²=49, 8²=64 → √50 is between 7 and 8, much closer to 7. Real-world applications: Geometry: hypotenuse c = √(a²+b²). Diagonal of 3×4 rectangle = √25 = 5. Diagonal of unit square = √2. - Physics: velocity from kinetic energy: v = √(2KE/m). RMS voltage = √(mean of squared voltages). - Statistics: standard deviation σ = √variance. For data [2,4,6,8]: variance=5, σ=√5≈2.236. - Finance: volatility and return calculations use square roots. Sharpe ratio involves √time for scaling.

Frequently Asked Questions

Only non-negative real numbers have real square roots. √16 = 4 and √0 = 0. The square root of a negative number (like √−9) is an imaginary number: √−9 = 3i, where i = √−1. Imaginary numbers are essential in electrical engineering, signal processing, and quantum mechanics. The calculator returns real results only — enter negative values if you need to work with imaginary numbers in complex number arithmetic.

Use the long division method or Newton's method for manual approximation. Long division: group digits in pairs from the decimal point, find the largest integer whose square fits each pair. Newton's method: start with any guess g, then iterate: g_new = (g + n/g) / 2. Example: √50, start with g=7. Next: (7 + 50/7)/2 = (7 + 7.143)/2 = 7.071. One more: (7.071 + 50/7.071)/2 ≈ 7.0711. Converges rapidly to √50 ≈ 7.07107.

A perfect square is an integer whose square root is also an integer. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144... produced by 1², 2², 3², 4²... To check if a number n is a perfect square: compute √n and verify the result is a whole number. Fast check: perfect squares only end in 0, 1, 4, 5, 6, or 9. A number ending in 2, 3, 7, or 8 is never a perfect square.

Simplify by factoring out perfect squares. √72 = √(36 × 2) = √36 × √2 = 6√2. Method: find the largest perfect square factor of the number under the root. 72 = 4 × 18 = 4 × 9 × 2 = 36 × 2. Pull out √36 = 6, leaving √2 under the radical. Always check for the largest perfect square factor, not just any — using 4 instead of 36 requires additional simplification steps.

Square roots and squares are inverse operations: √(n²) = n for n ≥ 0, and (√n)² = n for n ≥ 0. Example: √(7²) = √49 = 7, and (√49)² = 7² = 49. This inverse relationship is used to solve equations: x² = 81 → x = ±√81 = ±9 (both +9 and −9 satisfy x² = 81). Note: √(x²) = |x| (absolute value), not ±x, because the square root function returns only non-negative values.

The square root of a product equals the product of square roots: √(a×b) = √a × √b. Example: √(9×16) = √144 = 12, and √9 × √16 = 3 × 4 = 12 ✓. For fractions: √(a/b) = √a/√b. Example: √(25/49) = 5/7. These properties allow simplification of complex expressions. Important: √(a+b) ≠ √a + √b. Example: √(9+16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. The root of a sum is not the sum of roots.