Square Root Calculator
Find the square root of any non-negative number. The calculator shows the exact decimal value to your chosen precision, simplifies the radical when the number is not a perfect square, and — for perfect squares — identifies the integer result.
Enter a number
Enter a number and press Calculate square root.
What a square root is
The square root of a number n, written √n, is the
value that, when multiplied by itself, equals n. In other words:
√n = x ⟺ x² = n
For example, √9 = 3 because 3² = 9. Every positive
number has two square roots — one positive and one negative — but the symbol
√ (called the principal square root) refers to the non-negative
one. Zero has exactly one square root (zero itself), and negative numbers do
not have real square roots.
Perfect squares
Numbers like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144 are called perfect squares because their square roots are whole numbers. All other positive numbers have irrational square roots — infinite, non-repeating decimals that cannot be written exactly as a fraction.
Simplifying a square root
Many square roots can be written in a simpler form by pulling out perfect squares. The rule is:
√(a × b) = √a × √b
If one factor is a perfect square, its root comes outside the radical:
√72 = √(36 × 2) = 6√2
The calculator performs this simplification automatically when you enter an
integer, so you will see both the exact radical form (like 6√2)
and the decimal approximation.
Common square roots
| n | √n (decimal) | Exact / simplified |
|---|---|---|
| 1 | 1 | 1 (perfect square) |
| 2 | 1.41421356… | √2 (irrational) |
| 3 | 1.73205080… | √3 (irrational) |
| 4 | 2 | 2 (perfect square) |
| 8 | 2.82842712… | 2√2 |
| 9 | 3 | 3 (perfect square) |
| 16 | 4 | 4 (perfect square) |
| 25 | 5 | 5 (perfect square) |
| 50 | 7.07106781… | 5√2 |
| 72 | 8.48528137… | 6√2 |
| 100 | 10 | 10 (perfect square) |
| 144 | 12 | 12 (perfect square) |
Worked examples
Example 1 — Perfect square
Calculate √144.
12 × 12 = 144, so √144 = 12.
√144 = 12
Example 2 — Simplifying a radical
Calculate √72 in simplified form.
72 = 36 × 2, so √72 = √36 × √2 = 6√2 ≈ 8.4853.
√72 = 6√2 ≈ 8.485281374
Example 3 — Decimal input
Calculate √2.25.
1.5 × 1.5 = 2.25, so √2.25 = 1.5.
√2.25 = 1.5
Example 4 — Irrational result
Calculate √2.
√2 ≈ 1.4142135623730951… — an irrational number that cannot
be written exactly as a fraction.
√2 ≈ 1.414214 (rounded to 6 dp)
Common mistakes
-
Assuming √(a + b) = √a + √b.
This is false.
√(9 + 16) = √25 = 5, but√9 + √16 = 3 + 4 = 7. The square-root rule applies to multiplication, not addition. -
Forgetting that negative numbers have no real square root.
√(−4)is not−2— it is not a real number at all.(−2)² = 4, not −4. - Rounding too early. Round the final answer once. Chaining rounded intermediate values introduces small but cumulative errors.
-
Confusing the square root with "divide by 2".
The square root of n is the number whose square equals n. It is not
n ÷ 2, except in the special case
n = 0orn = 4(where√4 = 2 = 4/2).
Related calculators
Frequently asked questions
The square root of n is the number that, when multiplied by itself, equals n. For example, √25 = 5 because 5 × 5 = 25.
Because no real number multiplied by itself gives a negative result. (−3)² = 9, (+3)² = 9 — both positive. Negative square roots exist in the complex-number system, but they are not real numbers, so this calculator does not support them.
A simplified radical has no square factor left under the root sign. For example, √72 simplifies to 6√2 because 72 = 36 × 2 and 36 is a perfect square. The integer outside the radical (6) is called the coefficient.
No. Perfect squares (1, 4, 9, 16, 25, …) have integer square roots. All other positive integers have irrational square roots — infinite, non-repeating decimals. For example, √2 and √3 are irrational, but √9 and √16 are rational.
For everyday use, 4 decimal places are usually more than enough. For engineering or scientific work, 10–15 decimal places may be appropriate. The calculator supports up to 15 decimal places, matching the precision of double-precision floating-point arithmetic.