Square Root Calculator
Calculate square root, cube root, and various powers of numbers.
What is a square root?
Square root of x is the number that, when multiplied by itself, equals x. sqrt25 = 5 because 5 x 5 = 25. Every positive number has two roots: positive (principal) and negative. sqrt16 = +-4, but calculators show principal (positive) root. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Used in: Pythagorean theorem, standard deviation, quadratic formula, physics.
How do you calculate square root without a calculator?
Methods: 1) Memorize perfect squares (1-20^2). 2) Estimation: sqrt50 is between sqrt49=7 and sqrt64=8, closer to 7 (7.07). 3) Long division method (tedious). 4) Babylonian method: Guess, average with (number/guess), repeat. Example: sqrt10 -> Guess 3 -> (3 + 10/3)/2 = 3.167 -> Repeat -> 3.162. Most use calculators now, but estimation helps verify answers.
What is the difference between square root and cube root?
Square root (sqrt or x^1/2): Number multiplied by itself. ∛27 = 3 because 3 x 3 = 9. Cube root (∛ or x^1/3): Number multiplied by itself three times. ∛27 = 3 because 3 x 3 x 3 = 27. Higher roots: Fourth root ^4sqrt16 = 2 (2^4 = 16). Cube roots work for negatives: ∛(-8) = -2, but sqrt(-4) is imaginary (not real).
What are irrational square roots?
Most square roots are irrational (infinite non-repeating decimals). sqrt2 = 1.41421356... (never ends or repeats). Only perfect squares have rational roots. Common irrationals: sqrt2, sqrt3, sqrt5, PI, e. These appear everywhere in math and nature: sqrt2 in 45 degrees triangles, golden ratio involves sqrt5, circles use PI. Decimals are approximations - exact value is "sqrt2".