How to Find a Square Root
The square root of N is the number that gives N when squared. Perfect squares like 81 have exact whole-number roots (9); most numbers have irrational roots that go on forever — √2 ≈ 1.41421356… For whole numbers that aren't perfect squares, the root can still be simplified by pulling out perfect-square factors.
√(a² × b) = a√b
Simplify √72: the largest perfect square dividing 72 is 36, since 72 = 36 × 2. Therefore √72 = √36 × √2 = 6√2 ≈ 8.4853.
Perfect Squares Reference
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 9 | 81 |
| 2 | 4 | 10 | 100 |
| 3 | 9 | 11 | 121 |
| 4 | 16 | 12 | 144 |
| 5 | 25 | 13 | 169 |
| 6 | 36 | 14 | 196 |
| 7 | 49 | 15 | 225 |
| 8 | 64 | 16 | 256 |
Estimating Without a Calculator
To estimate √N, find the two perfect squares it sits between. For √50: it lies between √49 = 7 and √64 = 8, and much closer to 7 — a good estimate is 7.07. One refinement step, averaging 7 with 50 ÷ 7 ≈ 7.14, gives 7.071 — accurate to three decimal places. This is the ancient Babylonian method, and it's what calculators do internally, just repeated.
Frequently Asked Questions
What is a square root?
The square root of a number N is the value that, multiplied by itself, gives N. The square root of 49 is 7 because 7 × 7 = 49. Every positive number has two square roots (positive and negative); the √ symbol refers to the positive one.
How do you simplify a square root like √48?
Factor out the largest perfect square: 48 = 16 × 3, so √48 = √16 × √3 = 4√3. This calculator finds the simplified radical form automatically for whole numbers.
Can you take the square root of a negative number?
Not within the real numbers — no real number multiplied by itself gives a negative result. In advanced math, √−1 is defined as the imaginary unit i, so √−9 = 3i.
Which numbers have whole-number square roots?
Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on. Every other whole number has an irrational square root — an infinite, non-repeating decimal.
Cite This Calculator
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The CalculatorsGuide Editorial Team. (2026). Square Root Calculator. CalculatorsGuide. Retrieved from https://www.calculatorsguide.com/math/square-root-calculator/