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Nth Root Calculator
Use this free nth root calculator to find the square root, cube root, 4th root, 5th root, or any root of a number. Enter the number and the root, and it gives the answer to 5 decimal places, with a check that raises the answer back to the power to prove it. It also has a power mode, so you can compute x raised to any power, and it handles negative numbers correctly for odd roots.
The cube root of 27 is 3, because 3 × 3 × 3 = 27. Try n = 5 with x = 32 to get 2.
How it works
The nth root of a number x is the number that gives x when multiplied by itself n times. The cube root of 27 is 3, because 3 × 3 × 3 = 27. The 5th root of 32 is 2, because 2 × 2 × 2 × 2 × 2 = 32. The 4th root of 81 is 3, because 3 × 3 × 3 × 3 = 81. The square root is simply the 2nd root: the square root of 16 is 4, because 4 × 4 = 16. Every root has this same meaning, and the calculator finds it by raising x to the fractional power 1/n, which is the mathematical definition of the nth root.
Writing the root as a fractional exponent is the key to the whole tool. The nth root of x is x^(1/n), so the cube root of 27 is 27^(1/3), the 5th root of 32 is 32^(1/5), and the square root of 2 is 2^(1/2), which is 1.41421. That is how the calculator works internally, and it is also how you can check any answer on a scientific calculator that has a power button but no root button: raise x to the power 1 divided by n. The check shown under each result raises the answer back to the nth power, so for the cube root of 27 it shows 3³ = 27, proving the answer is right.
Some roots are whole numbers and some are not. The cube root of 27 is exactly 3, and the 5th root of 32 is exactly 2, but the square root of 2 is about 1.41421, an irrational number whose decimal never ends and never repeats. The calculator shows the answer to 5 decimal places by default, and you can switch to 2, 8, or 10 places. For irrational roots, more places are more accurate but never exact, which is why geometry often keeps the answer as √2 rather than writing out the decimal.
Negative numbers follow a simple rule. Odd roots of negative numbers are real: the cube root of −27 is −3, because (−3) × (−3) × (−3) = −27, and the 5th root of −32 is −2. Even roots of negative numbers have no real answer, because any real number raised to an even power is positive: no real number squared gives −4, and no real number to the 4th power gives −16. Those roots are imaginary numbers, written with i, and this calculator says so plainly instead of giving a wrong number. Zero is its own root for any n: the nth root of 0 is 0.
Roots show up everywhere once you know to look for them. The square root of 2 is the diagonal of a 1-by-1 square, which is why it appears in paper sizes and screen ratios. Cube roots appear when you know a volume and need the side of a cube: a box with a volume of 27 cubic inches has sides of 3 inches. In finance, the nth root finds average growth: if an investment grows from $1,000 to $1,331 in 3 years, the average yearly growth is the cube root of 1.331 minus 1, which is 10 percent a year. In statistics, the square root turns a variance back into a standard deviation in the original units.
∛27 = 27^(1/3) = 3
√2 = 2^(1/2) ≈ 1.41421
Check: (ⁿ√x)ⁿ = x
Common roots
| Root | Value |
|---|---|
| √2 | 1.41421 |
| √3 | 1.73205 |
| ∛27 | 3 |
| ∛64 | 4 |
| ⁴√81 | 3 |
| ⁵√32 | 2 |
| √144 | 12 |
| ∛1,000 | 10 |
Worked example
Find the cube root of 27, the 5th root of 32, the square root of 2, and the 4th root of 81.
Step 1: Cube root of 27 — 27^(1/3). 3 × 3 × 3 = 27, so the answer is 3.
Step 2: 5th root of 32 — 32^(1/5). 2 × 2 × 2 × 2 × 2 = 32, so the answer is 2.
Step 3: Square root of 2 — 2^(1/2) = 1.41421 to 5 decimal places. Check: 1.41421² ≈ 2.
Step 4: 4th root of 81 — 81^(1/4). 3 × 3 × 3 × 3 = 81, so the answer is 3.
Enter 27 with n = 3 in the calculator and it gives 3. Change to 32 with n = 5 and it gives 2. Change to 2 with n = 2 and it gives 1.41421. Change to 81 with n = 4 and it gives 3. Each answer raises back to the original number, which the calculator shows as its check.
FAQ
What is the nth root of a number?
The nth root of a number x is the number that gives x when multiplied by itself n times. The cube root of 27 is 3, because 3 × 3 × 3 = 27. The 5th root of 32 is 2, because 2 × 2 × 2 × 2 × 2 = 32. The square root is the 2nd root.
How do you calculate the cube root of 27?
Find the number that gives 27 when cubed. 3 × 3 × 3 = 27, so the cube root of 27 is 3. In general, the nth root of x is x raised to the power 1/n, so the cube root of 27 is 27^(1/3) = 3. This calculator does that power for any n.
What is the square root of 2?
The square root of 2 is about 1.41421. It is irrational, so its decimal never ends and never repeats. It is the length of the diagonal of a 1 by 1 square, by the Pythagorean theorem, and it appears in paper sizes, screen ratios and many geometry problems.
Can you take the root of a negative number?
Yes, for odd roots: the cube root of -27 is -3, because (-3) × (-3) × (-3) = -27. Even roots of negative numbers have no real answer, because any real number squared or raised to the 4th power is positive — the square root of -4 is imaginary, not a real number. This calculator gives the real odd root for negatives and explains when no real root exists.
What is the difference between a root and a power?
They are inverses. A power multiplies a number by itself: 2^5 = 32. A root undoes it: the 5th root of 32 is 2. In exponent form, the nth root of x is x^(1/n), so roots and powers are the same operation with a fractional exponent. This calculator has both modes, so you can check a root by raising it back to the power.