About the Cube Root and Nth Root Calculator
This cube root calculator finds the number that, multiplied by itself three times, gives your input — and it works for any other root too. Enter a number and a root index (3 for a cube root, 2 for a square root, 4 for a fourth root and so on) to get the decimal answer, a check that raises it back to the nth power, and whether the input is a perfect power.
For whole numbers it also writes the root in simplified radical form, pulling perfect factors outside the radical sign (∛54 = 3∛2, ⁴√48 = 2⁴√3), which is what algebra homework usually asks for. Volume problems are the classic use: the side of a cube holding 1,000 litres is ∛1 m³ = 1 m, and the edge of a 64 cm³ box is 4 cm.
Negative numbers have a real cube root (∛−8 = −2) and real odd roots in general, but even roots of negative numbers are not real numbers; the calculator flags those instead of returning an error.
With the default inputs, the root (decimal) is 3.7797631497. Change any value above to recalculate instantly.
How to use the cube root and nth root calculator
- 1Enter the number you want the root of.
- 2Set the root index: 3 for a cube root, or any whole number from 2 to 100.
- 3Read the decimal root and the check value that reverses it.
- 4Use the simplified radical form for exact algebra answers.
Formula and method
The nth root of x is the number r with rⁿ = x, which is the same as raising x to the power 1/n. The calculator uses the built-in cube root for n = 3 and otherwise a power followed by one Newton–Raphson refinement, so perfect powers such as ⁵√32 come out exactly 2.
To simplify a radical, the whole number is split into prime factors; every group of n identical primes moves outside the radical as a single factor, and what is left stays inside. Odd roots of negative numbers are negative, while even roots of negative numbers have no real value.
- x
- The radicand — the number you are taking the root of
- n
- Root index (3 for cube root)
- r
- The root, satisfying rⁿ = x
Worked examples
Cube root of 54
54 = 27 × 2 and 27 = 3³, so ∛54 = 3∛2 ≈ 3 × 1.259921 = 3.779763. It sits between the perfect cubes 27 and 64.
Perfect cube: ∛1728
12 × 12 × 12 = 1,728, so 1,728 is a perfect cube and its cube root is exactly 12.
Cube root of a negative number
(−5)³ = −125, so the real cube root of −125 is −5. Odd roots keep the sign of the number.
Fifth root of 100
100^(1/5) ≈ 2.511886, because 2.511886⁵ ≈ 100. 100 = 2² × 5² has no factor repeated five times, so the radical cannot be simplified.
Frequently asked questions
How do you calculate a cube root by hand?+
Find the perfect cubes on either side (for 50: 27 and 64, so the root is between 3 and 4), then refine with Newton’s method: r ← r − (r³ − x) ÷ (3r²). One step from 3.7 for x = 50 gives 3.6841, already very close to the true value 3.6840.
Can you take the cube root of a negative number?+
Yes. Because a negative number cubed is negative, every negative number has exactly one real cube root, which is negative: ∛−27 = −3. Only even roots (square, fourth…) of negatives are not real.
What are the perfect cubes from 1 to 1000?+
They are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000 — the cubes of 1 through 10. Knowing them makes it easy to estimate any cube root in that range.
How do I simplify a cube root?+
Factor the number and pull out any factor that is a perfect cube. For example 250 = 125 × 2, so ∛250 = 5∛2. For a fourth root pull out perfect fourth powers, and so on.
Is the cube root the same as raising to the power 1/3?+
Yes. ∛x and x^(1/3) are the same for positive x. On many calculators and in spreadsheets you can type x^(1/3), though some software returns an error for negative x even though a real cube root exists.