What this calculates
Enter a number and choose square root or cube root, and this gives the root as a decimal to eight places, the exact answer in simplest radical form — √72 = 6√2 — and whether the number is a perfect square or cube. Negative numbers work for cube roots; a negative square root has no real answer, and the calculator says so.
The formula
Square root
√x = r, where r × r = xCube root
∛x = r, where r × r × r = xSimplest radical form
if x = k² × m, then √x = k√m; if x = k³ × m, then ∛x = k∛m
k is taken as large as possible, so m has no perfect square (or cube) left inside it. When m comes out as 1, the number is a perfect square or cube and its root is a whole number.
Worked example
Cube roots work the same way: 54 = 27 × 2, so ∛54 = 3∛2, or about 3.77976315. A negative number keeps its sign: ∛−27 = −3, because −3 × −3 × −3 = −27.
When to use it, and the mistakes to avoid
Use it to simplify surds for maths homework, to find the side of a square from its area or of a cube from its volume, and to check whether a number is a perfect square or cube.
The mistakes that cost the most:
- Not taking out the largest square. 72 = 4 × 18 gives 2√18, which still simplifies; use 36 to get 6√2 in one step.
- Splitting a sum. √(9 + 16) is √25 = 5, not √9 + √16 = 7. Roots split over multiplication, never over addition.
- Halving instead of rooting. √64 is 8, not 32.
- Square-rooting a negative number. √−9 has no real answer; ∛−27 = −3 does.
- Rounding too early. Keep the radical form, like 6√2, until the last step and round only the final answer.
FAQ
What is a square root?
The square root of a number is the value that, multiplied by itself, gives that number. √144 = 12 because 12 × 12 = 144. Every positive number has two square roots, one positive and one negative; the √ sign means the positive one.
What is a cube root?
The cube root of a number is the value that, multiplied by itself three times, gives that number. ∛27 = 3 because 3 × 3 × 3 = 27. Unlike a square root, a negative number has a real cube root: ∛−27 = −3.
How do you simplify a square root?
Find the largest perfect square that divides the number, and take its root out in front of the sign. 72 = 36 × 2, and √36 = 6, so √72 = 6√2. Cube roots work the same way with perfect cubes: 54 = 27 × 2, so ∛54 = 3∛2.
Can you take the square root of a negative number?
Not as a real number, because any real number times itself is zero or positive. The answer is an imaginary number, √−9 = 3i, which this calculator does not cover. The cube root of a negative number is real, so choose cube root for those.
What is a perfect square?
A whole number whose square root is also a whole number: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 and so on. A perfect cube is the same idea for cube roots: 1, 8, 27, 64, 125, 216. The calculator tells you whether your number is one.
How accurate is the decimal answer?
It is shown to eight decimal places. Most square roots, like √2 = 1.41421356, never end or repeat, so any decimal is a rounded value; the simplest radical form, like √2 itself, is the exact answer.