Root Calculator: Square Root, Cube Root & Nth Root
Find any root in one step. The calculator gives the exact simplified form (like √72 = 6√2), the decimal value, and the working — so you learn the method, not just the answer.
What is a root?
A root undoes a power. If 5² = 25, then the square root of 25 is 5. In general, the nth root of a number x is the value r for which rⁿ = x. It can also be written as an exponent: ⁿ√x = x^(1/n).
The small number on the radical sign is the index. With no index, as in √x, it means 2 (the square root). The number under the sign is the radicand.
How to simplify a radical
To simplify ⁿ√x for a whole number x, split x into prime factors and pull out each group of n equal factors. Whatever cannot form a full group stays under the root.
Example: √72. 72 = 2 × 2 × 2 × 3 × 3. There is one pair of 2s and one pair of 3s, so 2 and 3 come out: 2 × 3 = 6. A single 2 is left inside, giving √72 = 6√2.
Example: ∛54. 54 = 2 × 3 × 3 × 3. One group of three 3s comes out, and 2 stays, giving ∛54 = 3∛2.
Roots of negative numbers
An odd root of a negative number is negative: ∛(−8) = −2, because (−2)³ = −8. An even root of a negative number has no real value, since no real number squared is negative. For the square root, the answer is imaginary: √−9 = 3i, where i = √−1.
Perfect squares and cubes, 1 to 15
| n | n² | n³ | n | n² | n³ |
|---|---|---|---|---|---|
| 1 | 1 | 1 | 9 | 81 | 729 |
| 2 | 4 | 8 | 10 | 100 | 1000 |
| 3 | 9 | 27 | 11 | 121 | 1331 |
| 4 | 16 | 64 | 12 | 144 | 1728 |
| 5 | 25 | 125 | 13 | 169 | 2197 |
| 6 | 36 | 216 | 14 | 196 | 2744 |
| 7 | 49 | 343 | 15 | 225 | 3375 |
| 8 | 64 | 512 | Memorise these for exams | ||
Root calculator FAQ
The square root of x is the value that, multiplied by itself, gives x. For example, √144 = 12 because 12 × 12 = 144. Enter the number above, keep the root index at 2, and the calculator shows the exact and decimal answer.
A square root (index 2) asks which number multiplied by itself gives x. A cube root (index 3) asks which number multiplied by itself three times gives x. For example, √81 = 9 and ∛27 = 3.
Factorise the number into primes and pull out every group that matches the root index. 72 = 2³ × 3², so √72 = 3 × 2 × √2 = 6√2. The calculator does this automatically for whole numbers.
Not within the real numbers. An even root of a negative number has no real value. The square root of −9 is the imaginary number 3i. Odd roots such as the cube root are fine: ∛(−27) = −3.
A perfect square is the square of a whole number (1, 4, 9, 16, 25 …). A perfect cube is the cube of a whole number (1, 8, 27, 64, 125 …). Their roots are whole numbers.
The nth root of x equals x raised to the power 1/n. For example, the 4th root of 81 is 81^(1/4) = 3. Choose "Nth root" above, set the index to n, and enter x.