Square Root Calculator
Calculate the square root, cube root, or any nth root of a number. Enter a value to get the principal root, the simplified radical form, whether it is a perfect power, and a full step-by-step breakdown. Use reverse-solve mode to find the number that produces a given root.
Formula
Worked example
Square root: for n = 144, find r such that r^2 = 144. Since 12^2 = 144, the answer is 12, a perfect square. For n = 72: factor out the largest perfect square (36), so √72 = 6√2 ≈ 8.485. Cube root example: the cube root of 27 is 3 (perfect cube), and the cube root of 30 ≈ 3.107 (irrational). Reverse-solve: if the square root is 7, the original number is 7^2 = 49.
What a square root is and how to read the symbol
The square root of a number n is a value r such that r × r = n. The radical symbol √ most likely comes from the Latin word "radix" (root), and the horizontal bar (called the vinculum) means "bond" in Latin. Because both a positive number and its negative counterpart square to the same result, every positive number technically has two square roots, but the principal root is always the non-negative one, which is what the radical symbol conventionally denotes. This calculator returns the principal root.
nth roots: cube roots, fourth roots, and beyond
The square root is a special case of the nth root. The cube root (ⁱ³√x) finds a value r such that r^3 = x, and it works for negative numbers since (-2)^3 = -8. The fourth root and beyond follow the same pattern: the nth root of x is x^(1/n). Even-index roots (square, fourth, sixth, and so on) of negative numbers are imaginary and have no real value, while odd-index roots (cube, fifth, seventh, and so on) of negative numbers do have a real answer. Use the root index input to switch between any of these modes.
Simplifying radicals
A radical can often be written in a simpler exact form by factoring out the largest perfect-square factor from the radicand. For example, √72 = √(36 × 2) = 6√2, because 36 is a perfect square and √36 = 6. The number outside the radical (6) is called the coefficient and the number inside (√2) is the simplified radicand. This exact form is preferred in algebra because it avoids rounding. The calculator shows this simplified form for integer square roots automatically.
Arithmetic with square roots
Square roots obey useful arithmetic rules. Multiplication: √(x × y) = √x × √y, so √(4 × 9) = 2 × 3 = 6. Division: √(x/y) = √x / √y. Addition and subtraction only work when the radicands match: 3√2 + 5√2 = 8√2, but √2 + √3 cannot be combined further. These rules are helpful when you need to simplify an expression before plugging into a calculator.
Reverse-solve mode: from root to number
Sometimes you know the root and need to find the original number. Switch to "Root to number (reverse-solve)" mode, enter the root value and the index, and the calculator computes root^n. This is the squaring (or nth-power) operation in reverse. For example, if the square root is 7.5, the original number is 7.5^2 = 56.25. This is useful when checking work, scaling areas, or solving equations of the form √x = k.
Perfect squares and irrational roots
A perfect square is a non-negative integer whose square root is also a whole number: 1, 4, 9, 16, 25, 36, and so on. When a number is not a perfect square, its square root is irrational, meaning it is a non-terminating, non-repeating decimal like √2 ≈ 1.41421356. The same idea extends to higher powers: a perfect cube is a number like 8, 27, or 64 whose cube root is a whole number. The calculator flags whether your number is a perfect power for the chosen root index.
Why negative numbers under even roots have no real answer
Any real number raised to an even power (2, 4, 6, and so on) produces a non-negative result, because a positive times a positive is positive and a negative times a negative is also positive. There is therefore no real number whose square (or fourth power, and so on) is negative. Mathematicians extend the number system using the imaginary unit i, defined so that i^2 = -1. The square root of -9 is then 3i, the square root of -25 is 5i, and so on. These are complex numbers, not real numbers. Odd-index roots of negative numbers are real: the cube root of -8 is -2, because (-2)^3 = -8.
Perfect squares and their square roots (1 to 225)
| Number (n) | Square root (√n) | Perfect square? |
|---|---|---|
| 1 | 1 | Yes |
| 2 | √2 ≈ 1.4142 | No |
| 3 | √3 ≈ 1.7321 | No |
| 4 | 2 | Yes |
| 5 | √5 ≈ 2.2361 | No |
| 8 | 2√2 ≈ 2.8284 | No |
| 9 | 3 | Yes |
| 12 | 2√3 ≈ 3.4641 | No |
| 16 | 4 | Yes |
| 18 | 3√2 ≈ 4.2426 | No |
| 25 | 5 | Yes |
| 36 | 6 | Yes |
| 49 | 7 | Yes |
| 50 | 5√2 ≈ 7.0711 | No |
| 64 | 8 | Yes |
| 72 | 6√2 ≈ 8.4853 | No |
| 81 | 9 | Yes |
| 100 | 10 | Yes |
| 121 | 11 | Yes |
| 144 | 12 | Yes |
| 169 | 13 | Yes |
| 196 | 14 | Yes |
| 225 | 15 | Yes |
Any integer between two of these values has an irrational square root. Values in simplified radical form are shown where the root is not a whole number.
Frequently asked questions
What is the square root of a negative number?
A negative number has no real square root. Its square root is an imaginary number, written as a multiple of i (the imaginary unit, defined as √(-1)). For example, √(-4) = 2i. This calculator reports the imaginary form for negative inputs under even-index roots and flags the result as imaginary.
Does every positive number have two square roots?
Yes. Every positive number has a positive and a negative square root, since both square to the same value. For example, both 5 and -5 are square roots of 25. By convention the radical symbol √ refers to the principal (non-negative) root, which is what this calculator displays. Zero has exactly one square root: itself.
How do I simplify a square root like √72?
Factor out the largest perfect-square factor from the radicand. For 72: the largest perfect-square factor is 36, so √72 = √(36 × 2) = √36 × √2 = 6√2. The calculator does this automatically for integer inputs in square root mode and shows the result in the Simplified radical field.
What is the cube root, and when does it give a real answer for negatives?
The cube root of x is the number r such that r^3 = x. Unlike square roots, cube roots of negative numbers are real: the cube root of -8 is -2, because (-2)^3 = -8. In general, odd-index roots always produce real answers for any real input, while even-index roots of negative numbers are imaginary. Set the root index to 3 (or any odd number) to calculate roots of negative values.
How does reverse-solve mode work?
Switch to "Root to number (reverse-solve)" mode, enter the root value you know and the root index. The calculator raises your value to the nth power to find the original number. For instance, if you know the square root is 8.5, it computes 8.5^2 = 72.25. This is useful when you are working backward from an answer or checking whether your root estimate is correct.
Why does squaring the displayed result not always match the original number exactly?
For perfect squares (and perfect nth powers) the check matches exactly. For other numbers the root is irrational, meaning it cannot be expressed with a finite number of digits. The displayed value is rounded, so squaring it back can differ from the original by a tiny rounding amount, typically in the last decimal place.
Can I multiply or add square roots together?
Multiplication is straightforward: √x × √y = √(xy). Division works similarly: √x / √y = √(x/y). Addition and subtraction only work when the radicands are identical: 3√5 + 7√5 = 10√5, but √3 + √5 cannot be simplified further. When radicands differ, simplify each radical first and see whether like terms emerge.