The cube root of a number is the value that, multiplied by itself three times, gives that number. The cube root of 27 is 3, because 3 × 3 × 3 = 27. Unlike square roots, cube roots work for negative numbers too: the cube root of −8 is −2, since three negatives multiplied together stay negative. Every real number has exactly one real cube root.

Can you take the square root of a negative number?

Not within real numbers. Squaring any real number gives a positive result, since negative times negative is positive, so nothing real squares to −9. Mathematicians handle this by defining i as the square root of −1, making √(−9) equal to 3 i . This calculator works in real numbers only, so it reports even roots of negatives as undefined. Odd roots of negatives are fine.

The nth root reverses raising to the nth power. The fourth root of 81 is 3, because 3⁴ = 81. You can write any root as a fractional power — the nth root of x equals x 1/n — which makes algebra much simpler. Odd roots work for any number, positive or negative; even roots only work for zero and positive numbers.

What makes a number a perfect square?

A perfect square is a whole number multiplied by itself, so its square root comes out exactly. 16, 25, 36, 49 and 64 are perfect squares; 20 and 30 are not. Recognising them tells you instantly whether an answer will be whole or a decimal. The same idea applies to perfect cubes such as 8, 27, 64 and 125, whose cube roots are whole numbers.

How do I simplify a square root?

Split the number into the largest perfect square times another number, then take the perfect square's root outside. √72 = √(36 × 2) = 6√2.

Square Root Calculator — Square, Cube & Nth Roots

About the Roots Calculator

This calculator handles square roots, cube roots and any nth root you ask for. Enter the number, set the degree, and you get the result along with a check — the answer raised back to that power, so you can see it returns your original number.

It does one thing many calculators refuse to do: odd roots of negative numbers. The cube root of −8 is −2, which is a perfectly real answer, but a naive implementation returns an error because it reaches for a power function that cannot cope. This tool gets it right, and explains why even roots of negatives genuinely have no real answer.

It also tells you whether your number is a perfect square or perfect cube, which is the quickest way to know if an answer will come out whole before you start.

How to use the Roots Calculator

Enter the number.
Choose the root, or pick Custom for anything else.
Select Calculate root.
The check column raises the result back to that power so you can confirm it.
Negative numbers work with odd roots only.

What a root actually is (sqrt calculator basics)

An nth root reverses raising to the nth power. If you cube 3 and get 27, then the cube root of 27 is 3. It is the same relationship as multiplication and division, one step up.

√x = the number that, multiplied by itself, gives x

∛x = the number that, cubed, gives x

ⁿ√x = x^1/n — a root is just a fractional power

Worked example

Find the cube root of −125.

Ask what number cubed gives −125. Since (−5) × (−5) × (−5) = −125, the answer is −5.

∛(−125) = −5, and it is exact

Three negatives multiplied together stay negative, which is why odd roots of negative numbers work. Two negatives cancel to a positive, which is why even roots of negatives do not.

Why even roots of negatives have no real answer

Squaring any real number gives a positive result. Positive times positive is positive; negative times negative is also positive. So no real number squared can produce −9, and √(−9) has no real solution.

Mathematicians extend the number system to handle this, defining i as the square root of −1. Then √(−9) = 3i, an imaginary number. This calculator works in real numbers, so it reports even roots of negatives as undefined rather than inventing a result.

Root	Of a positive	Of a negative
Square (2nd)	√16 = 4	No real answer
Cube (3rd)	∛27 = 3	∛(−27) = −3
Fourth	∜81 = 3	No real answer
Fifth	√[5]32 = 2	−32 gives −2

The pattern is simple: odd roots work for any number, even roots only for zero and positives.

Perfect squares and perfect cubes

A perfect square is a whole number times itself. Recognising them saves time, and they show up constantly in exam questions.

n	n² (perfect square)	n³ (perfect cube)
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	729
10	100	1,000
12	144	1,728

Roots as fractional powers

Writing a root as a power makes the algebra much easier, because the ordinary power rules then apply.

ⁿ√x = x^1/n — the fifth root of 32 is 32^0.2 = 2.
ⁿ√(x^m) = x^m/n — the cube root of 8² is 8^2/3 = 4.
√a × √b = √(ab) — which is how radicals get simplified.
1 ÷ ⁿ√x = x^−1/n — a negative exponent flips the fraction.

Roots turn up throughout the rest of these tools — the kinetic energy calculator takes a square root to solve for velocity, and the area calculator uses one in Heron’s formula to find a triangle’s area from three sides.

Every positive number technically has two square roots: 4 and −4 both square to 16. The radical symbol √ conventionally means the positive one, called the principal root, and that is what this calculator returns. When solving an equation like x² = 16, remember to include both.

Simplify radicals calculator: simplest radical form

Find the largest perfect square that divides the number. Take its root outside.

Write the number as (perfect square) × (something else).
Take the square root of the perfect square.
Leave the rest under the root sign.

Root	Split	Simplified
√12	√(4 × 3)	2√3
√50	√(25 × 2)	5√2
√72	√(36 × 2)	6√2
√180	√(36 × 5)	6√5

How to estimate a square root by hand

Find the two perfect squares either side, then guess between them. √50 is between √49 = 7 and √64 = 8, and very close to 49. So it is just over 7 — about 7.07. The calculator above gives the exact decimal.

Quadratic root finder: roots of a quadratic equation

The "roots" of ax² + bx + c = 0 are the x values that make it true. Use the quadratic formula:

x = (−b ± √(b² − 4ac)) ÷ 2a

For x² − 5x + 6 = 0: b² − 4ac = 25 − 24 = 1. x = (5 ± 1) ÷ 2, so x = 3 or x = 2. If b² − 4ac is negative, there are no real roots.

Other ways to use this calculator

Square, cube and higher roots, plus a few related tasks.

Nth root and fourth root calculator — choose Custom degree for any root.
Cube and square root calculator — pick the root from the menu; negative numbers work for odd roots.
Square root simplifier — use the method above to write roots like √72 as 6√2.
Roots of a quadratic equation — use the quadratic formula above.
Cube root calculator and radical calculator — choose the root index: 2, 3 or any n.
Simplify square root — see the simplest radical form section above.
Roots of quadratic equation — use the quadratic formula section above.

Saved calculators

Translate this page

Roots Calculator