Square Root Calculator
Enter a positive number and the calculator returns its square root in decimal form to 15 digits and, where possible, its exact simplified radical form — √72 becomes 6√2, √200 becomes 10√2. For perfect squares you get an integer; for negatives you get i notation with the imaginary unit pulled out.
How the root is computed
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1
Enter the radicand
The number under the radical. Positive, negative or zero.
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2
Decimal form
Computed via the IEEE 754 square root instruction — accurate to 15 significant digits.
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3
Simplified radical form
Factor out perfect-square divisors. √72 = √(36 × 2) = 6√2.
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4
Show the working
Step-by-step factorisation is displayed so you can reproduce it by hand.
Perfect squares to know
| n | n² | √(n²) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 10 | 100 | 10 |
| 11 | 121 | 11 |
| 12 | 144 | 12 |
| 13 | 169 | 13 |
| 14 | 196 | 14 |
| 15 | 225 | 15 |
| 16 | 256 | 16 |
| 25 | 625 | 25 |
Simplifying non-perfect squares
The trick is to find the largest perfect-square factor:
- √50 = √(25 × 2) = 5√2
- √72 = √(36 × 2) = 6√2
- √108 = √(36 × 3) = 6√3
- √500 = √(100 × 5) = 10√5
- √1000 = √(100 × 10) = 10√10
If the result still has a non-square factor, repeat: √180 = √(36 × 5) = 6√5, not √(4 × 45) = 2√45 (not fully simplified).
Common decimal values
- √2 ≈ 1.41421 (Pythagoras in a unit square)
- √3 ≈ 1.73205 (diagonal of a cube)
- √5 ≈ 2.23607 (appears in the golden ratio (1+√5)/2)
- √7 ≈ 2.64575
- √π ≈ 1.77245 (used in statistics, Gaussian integrals)
- √10 ≈ 3.16228
- √(1000) ≈ 31.6228 (each 10x increases √ by ~3.16x)
Negative numbers and imaginaries
The square root of a negative number is not defined in the reals. In the complex numbers, √(−x) = i√x for positive x. So √(−4) = 2i. The calculator reports the imaginary form rather than a decimal for negative inputs.
Square root vs nth root
The calculator handles square (2nd) roots. For cube roots, fourth roots, etc., use a general nth-root tool. Key identities:
- √(ab) = √a × √b (only if a and b are non-negative)
- √(a/b) = √a / √b (only if b > 0)
- (√a)² = a (only if a >= 0)
History pointer
The radical symbol √ evolved from the letter r (for radix, root in Latin) in the 1500s. The horizontal bar (the vinculum) was added in the 17th century to delimit what is under the root.
Frequently Asked Questions
Every positive number has two square roots: +x and −x. The principal root (the non-negative one) is what √ usually refers to. Quadratic equations use both.
Just 5, by convention. √ returns the principal (non-negative) root. When solving x² = 25, both 5 and −5 satisfy the equation, so you write x = ±5.
Historical methods: digit-by-digit long division algorithm, Newton’s method (iterative: x_new = (x + a/x)/2), or factor-and-simplify for roots of perfect-square-rich numbers. Newton’s method converges quickly — three iterations gives 10-digit accuracy for most inputs.
Proven by the Greeks via contradiction: if √2 = p/q in lowest terms, then 2q² = p², making p even, so p = 2k, then 2q² = 4k², giving q² = 2k², making q even too — contradicting lowest terms. So √2 cannot be a fraction; it is irrational.
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