Root Calculator: Square, Cube and n-th Roots
Enter a number and the root index to get its square, cube or n-th root to ten decimal places, whether the root is exact, and its simplified radical form (√72 = 6√2).
- As a power72^(1/2)
- Simplified radical6√2
- Exact root?No, it never ends (irrational)
- Check: root²72
How it is calculated
The n-th root of a number is the number whose n-th power gives it: ∛27 = 3 because 3³ = 27. Square roots (n = 2) and cube roots (n = 3) are the most common. Taking a root is the same as raising to the power 1/n. The root of a whole number that is not a perfect square (or cube) is irrational: its decimals never end or repeat, so the result is rounded to ten places. Simplifying moves the n-th powers inside the number out of the root, so √72 is written 6√2. Negative numbers have odd roots (∛−27 = −3) but no real even roots.
ⁿ√x = x^(1/n) · ⁿ√(kⁿ × m) = k × ⁿ√mx: the number · n: root index (2 for square, 3 for cube root) · k: the factor taken out of the root · m: what stays under it.- Factor the number into primes: 72 = 2³ × 3².
- Each group of n equal factors comes out as one factor: for a square root 2² → 2 and 3² → 3, so 2 × 3 = 6 outside.
- Factors left over stay under the root: one 2. So √72 = 6√2.
- As a decimal: 6 × √2 ≈ 6 × 1.4142135624 ≈ 8.4852813742.
Example
√72 = 6√2 ≈ 8.4852813742. ∛54 = ∛(3³ × 2) = 3∛2 ≈ 3.7797631497. ∜81 = 3, because 3⁴ = 81. √0.25 = 0.5, because 0.5 × 0.5 = 0.25.
Frequently asked questions
Does a negative number have a square root?
Not among real numbers, since no real number squared is negative. In complex numbers √−1 = i, so √−16 = 4i. Odd roots of negative numbers are real: ∛−8 = −2.
Why is √2 irrational?
If √2 = p/q in lowest terms, then p² = 2q², which forces p and then q to be even, contradicting lowest terms. So √2 ≈ 1.4142135624… is not any fraction.
How do I work out a square root by hand?
By Newton’s method: start from a guess and replace it with (guess + number ÷ guess) ÷ 2. For √144 from 10: 12.2 → 12.0016 → 12.0000001; each step roughly doubles the correct digits.
Which numbers are perfect squares?
Squares of whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 … Their square roots are whole numbers, and the calculator shows “Exact root: Yes”.
Why simplify a radical?
It makes radicals easy to add and compare: √50 + √18 = 5√2 + 3√2 = 8√2. Exams usually want answers in simplified form.