Square Root Calculator — √n and Simplified Radical Form
The square root of a number n is the value that, multiplied by itself, gives n. This calculator returns the answer two ways: as a decimal value (for example √72 ≈ 8.485) and, for whole numbers, as a simplified radical or surd (for example √72 = 6√2). You can also switch the root degree to find cube roots and any nth root.
Calculate a square root
How to calculate a square root
To find the square root of a number n, you look for a value x such that x × x = n. Every positive number actually has two square roots — one positive and one negative — but the symbol √ refers to the principal (positive) root. So √25 = 5, because 5 × 5 = 25.
For perfect squares (numbers like 1, 4, 9, 16, 25…) the root is a whole number. For every other positive integer the square root is irrational: its decimal expansion never ends and never repeats. In that case we either round the decimal (for example √2 ≈ 1.4142136) or keep the exact value in radical form (√2).
By hand, a quick method is estimation and averaging (the Babylonian method): guess a value, divide n by your guess, then average the two. Repeat, and the result converges on the true square root. For example, to estimate √72: 72 ÷ 8 = 9, average of 8 and 9 is 8.5; 72 ÷ 8.5 ≈ 8.47, average ≈ 8.485 — already accurate to three decimal places.
Simplifying radicals
A radical is simplified when no perfect-square factor (other than 1) remains under the root sign. The trick is to split n into a perfect square times whatever is left over, then take the square root of the perfect-square part outside the sign.
Take √72 as the worked example:
- Find the largest perfect square that divides 72. That is 36 (= 6²), because 72 = 36 × 2.
- Rewrite: √72 = √(36 × 2) = √36 × √2.
- √36 = 6, so √72 = 6√2.
Check it back: 6√2 ≈ 6 × 1.41421 ≈ 8.485, which matches the decimal value. A few more worked examples:
- √50 = √(25 × 2) = 5√2
- √48 = √(16 × 3) = 4√3
- √98 = √(49 × 2) = 7√2
- √12 = √(4 × 3) = 2√3
- √45 = √(9 × 5) = 3√5
If a number has no perfect-square factor bigger than 1 (for example √13 or √15), it is already in simplest radical form and cannot be reduced further.
Perfect squares reference table
Memorising the first fifteen perfect squares makes it far quicker to spot factors when simplifying radicals.
| n | n² (perfect square) | √(n²) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 7 | 49 | 7 |
| 8 | 64 | 8 |
| 9 | 81 | 9 |
| 10 | 100 | 10 |
| 11 | 121 | 11 |
| 12 | 144 | 12 |
| 13 | 169 | 13 |
| 14 | 196 | 14 |
| 15 | 225 | 15 |
Frequently asked questions
What is the square root of 72?
The square root of 72 is approximately 8.485. In exact (simplified radical) form it is 6√2, because 72 = 36 × 2 and √36 = 6. Since 72 is not a perfect square, the decimal value is irrational and never terminates.
What is a simplified radical or surd?
A simplified radical (called a "surd" in the UK) is a square root written in its most reduced exact form, with no perfect-square factor left under the root sign. For example √50 simplifies to 5√2. It keeps the value exact rather than rounding to a decimal.
Can you take the square root of a negative number?
Not within the real numbers. A negative number has no real square root because no real value squared gives a negative result. The answer is an imaginary number — for example √(−9) = 3i, where i is the imaginary unit √(−1). Odd roots (such as a cube root) of negative numbers do have real values.
What is the difference between a square root and an nth root?
A square root asks "what number times itself gives n?" An nth root asks "what number raised to the power n gives the original value?" Setting the root degree to 3 gives a cube root (∛), to 4 gives a fourth root, and so on. A square root is simply the nth root with n = 2.
How do I know if a number is a perfect square?
A number is a perfect square if its square root is a whole number — like 1, 4, 9, 16, 25, 36 and so on. A quick test: take the square root; if there is no fractional part, it is a perfect square. This calculator tells you directly whether the number you entered is a perfect square.
Related Calculators
- Scientific Calculator — roots, powers, trig and more.
- Exponent Calculator — powers, negative and fractional exponents.
- Percentage Calculator — percentages, increases and decreases.
Last reviewed: June 2026