平方根计算器
求平方根、立方根和n次方根
calculators.square-root.perfectSquares
Root Properties
来源与方法
How to Use the Square Root Calculator
The square root calculator finds square roots, cube roots, and nth roots of any number. Get both the decimal answer and simplified radical form, plus learn the prime factorization method step-by-step.
What is a Square Root?
The square root of a number is a value that, when multiplied by itself, gives the original number:
√25 = 5 because 5 × 5 = 25
The symbol √ is called the radical sign.
Simplifying Square Roots
To simplify a square root, find perfect square factors:
- Factor the number into prime factors
- Pair up matching primes
- Bring pairs outside the radical
Example: √72 = √(36 × 2) = √36 × √2 = 6√2
Perfect Squares
Numbers whose square roots are whole numbers:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225...
Cube Roots and Higher
Cube roots (³√) find what number multiplied by itself three times gives the original:
³√27 = 3 because 3 × 3 × 3 = 27
The nth root (ⁿ√) extends this to any power.
Square Roots of Negative Numbers
The square root of a negative number is an imaginary number, denoted with i:
√(-1) = i, so √(-9) = 3i
Related Calculators
For powers and exponents, use our exponent calculator. For logarithms, try our log calculator.
常见问题
Factor the number into perfect squares times a remainder. √72 = √(36 × 2) = √36 × √2 = 6√2. Find the largest perfect square factor. Prime factorization helps: 72 = 2³ × 3² = (2 × 3)² × 2 = 36 × 2.
The square root of a negative number is imaginary, denoted with i where i = √(-1). So √(-9) = √(9 × -1) = √9 × √(-1) = 3i. Imaginary numbers are used extensively in engineering and physics.
A square root asks "what times itself equals this?" A cube root asks "what times itself three times equals this?" ³√27 = 3 because 3×3×3 = 27. Unlike square roots, cube roots of negative numbers are real: ³√(-8) = -2.
Perfect squares have whole number roots: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144... These are squares of integers: 1², 2², 3², etc. Most numbers have irrational square roots that go on forever without repeating.
Use estimation and refinement. For √50: 7² = 49 (too small), 8² = 64 (too big), so √50 is between 7 and 8. Try 7.1: 7.1² = 50.41 (close!). Refine to 7.07² = 49.98. Continue until desired precision is reached.
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