Square Root Calculator
This square root calculator takes any root: enter the number and the index – 2 for a square root, 3 for a cube root. You get the answer, the simplified radical and the working.
Example: Number under the root (radicand) 72 · Index n 2 → Result: √72 ≈ 8.48528137424. Source: Wolfram MathWorld – Square root. Updated: .
How it is calculated
The square root √a is the non-negative number that gives a when multiplied by itself: √144 = 12 because 12 × 12 = 144. More generally, the nth root ⁿ√a is the number whose nth power is a. If a is not a perfect square the root is irrational – √2 ≈ 1.41421356 – and the calculator shows it rounded, simplified and approximated step by step.
Simplifying square roots
To simplify a radical, split the radicand into prime factors and move every factor that appears often enough outside the root. Example: 72 = 2³ × 3² = 6² × 2, so √72 = 6√2. For cube roots you need groups of three: 54 = 3³ × 2, so ∛54 = 3∛2. The calculator shows exactly this factorisation.
Finding a square root by hand: the Babylonian method
The Babylonian (Heron’s) method improves a guess x with xnew = (x + a ÷ x) ÷ 2. Each step roughly doubles the number of correct digits. For √2: 1 → 1.5 → 1.41667 → 1.4142157 → 1.41421356. For higher roots the calculator uses Newton’s method xnew = ((n − 1)x + a ÷ xⁿ⁻¹) ÷ n.
Rules for roots
- √(ab) = √a × √b and √(a ÷ b) = √a ÷ √b (for a, b ≥ 0)
- ⁿ√a = a^(1/n), so √a = a^0.5
- √(a²) = |a| – a square root is never negative
- Negative numbers only have odd real roots: ∛−27 = −3; √−4 is not a real number.
Powers and logs are handled by the exponent calculator, full expressions with roots by the scientific calculator, and roots in right triangles by the Pythagorean theorem calculator.
Frequently asked questions
How do you find a square root without a calculator?
Use the Babylonian method: guess x, then repeat x = (x + a ÷ x) ÷ 2. For √10 starting at 3: 3 → 3.1667 → 3.16228 – five correct digits after three steps.
How do you simplify a square root?
Write the radicand as a perfect square times the rest and take the square outside: √50 = √(25 × 2) = 5√2 and √72 = √(36 × 2) = 6√2.
What is the square root of a negative number?
Even roots of negative numbers are not real numbers (they are imaginary, √−4 = 2i). Odd roots are real: ∛−8 = −2 because (−2)³ = −8.
How do you calculate a cube root?
Enter 3 as the index. ∛27 = 3 because 3 × 3 × 3 = 27. Cube roots that do not come out evenly, like ∛54 = 3∛2 ≈ 3.78, are simplified and approximated.
Is a square root the same as raising to the power 0.5?
Yes: √a = a^(1/2) = a^0.5, and in general ⁿ√a = a^(1/n). That is how roots can be done with the power key.
Sources and legal basis
- Wolfram MathWorld – Square root
- Wolfram MathWorld – nth root
- Wolfram MathWorld – Newton’s iteration (Heron’s method)
- DLMF §4.2: NIST Digital Library of Mathematical Functions – powers and roots
As of:
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