Square Root & Cube Root Calculator
Roots to any precision, simplified radicals with steps, and every complex root.
Root
This is the last answer worked out. Fix the input above to update it.
Decimal value
Simplest radical form
Perfect powers
All roots
| k | Root | Angle | Note |
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Long division method
| Step | Bring down | Number | Divisor | × digit | Subtract | Remainder | Root so far |
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About the Square Root & Cube Root Calculator
Type a number to get its square root, cube root or any nth root (n up to 100) to as many as 1,000 decimal places. Real roots are worked out with exact whole-number arithmetic, so every digit shown is correct and only the last one is rounded. The calculator also writes the root in simplest radical form — √72 = 6√2, ∛54 = 3∛2, √(1/2) = √2/2 — showing the prime factorisation and which factors come out of the root, and checks whether the number is a perfect square or cube.
Negative and complex numbers work too: √−72 = 6i√2, and the table lists all n complex nth roots, such as the three cube roots of 8, or √(3 + 4i) = 2 + i. For square roots, the digit-by-digit long division method is worked out step by step, the way it is taught in school.
How to use it
- Type the number under the root sign: a whole number, a decimal, a fraction such as
8/27, a negative number or a complex number such as3+4i. You can also type√72or∛54. - Choose √ Square, ∛ Cube or ⁿ√ Other and enter the index n.
- Set the number of decimal places, from 0 to 1,000.
- Read the root, its simplest radical form with the steps, the perfect-power checks, the list of all the roots and, for square roots, the long division working. Copy the value or the simplest form.
Examples
√72
6√2 ≈ 8.4852813742
72 = 2³ × 3² = 6² × 2, so √72 = 6√2.
∛54
3∛2 ≈ 3.7797631497
54 = 2 × 3³, so the 3³ comes out as 3.
√(1/2)
√2/2 ≈ 0.7071067812
√−72
6i√2
A negative number has no real square root; i = √−1.
√(3 + 4i)
2 + i and −2 − i
Check: (2 + i)² = 4 + 4i + i² = 3 + 4i.
∛8
2, −1 + 1.7320508076i and −1 − 1.7320508076i
Common uses
- Checking homework on surds, radicals and simplest radical form.
- Getting square roots and cube roots to many decimal places.
- Finding the complex roots of a number for algebra, engineering or signal-processing work.
Simplest radical form
To simplify ⁿ√N, write N as a product of primes and take out every group of n equal factors. √72 = √(2 × 2 × 2 × 3 × 3): the pair of 2s and the pair of 3s come out as 2 × 3 = 6, and one 2 stays inside, so √72 = 6√2. A fraction is simplified by making its denominator a perfect power first (rationalising): √(1/2) = √(2/4) = √2/2. When every exponent left inside shares a factor with the index, the index itself gets smaller: ⁶√8 = ⁶√(2³) = √2.
Square roots by long division
- Split the digits into pairs, starting at the decimal point: 72 becomes 72 . 00 00.
- Find the largest digit whose square fits into the first pair: 8² = 64 ≤ 72. Write 8 in the root and subtract: 72 − 64 = 8.
- Bring down the next pair to make 800. Double the root so far (8 × 2 = 16) and find the largest digit d with 16d × d ≤ 800: 164 × 4 = 656. Write 4 in the root and subtract: 800 − 656 = 144.
- Repeat: bring down 00 to make 14400, double 84 to 168, and 1688 × 8 = 13504, so the next digit is 8 — √72 ≈ 8.48.
Negative and complex numbers
Even roots of negative numbers are not real: √−72 = √72 × √−1 = 6√2 × i. Odd roots keep the sign: ∛−8 = −2. Every non-zero number, real or complex, has exactly n different nth roots in the complex numbers. They all have the same size, |z|^(1/n), and are spaced 360°/n apart around a circle, starting from the principal root at angle θ/n, where θ is the angle of z. The table lists all of them and marks the principal root and the real ones.
Limitations
- Real roots: up to 1,000 decimal places, with n × (places + 1) at most 20,000 — so a 100th root can have up to 199 places. Complex roots: up to 50 decimal places, and at most 100 roots are listed.
- The simplest radical form needs the prime factors of the number. If a very large number cannot be factorised within a fraction of a second, it is simplified as far as possible and the steps say so. A fraction is not rationalised when that would put a number of more than 60 digits under the root — ¹⁰⁰√(1/7), for example, stays as it is.
- The long division working is shown for square roots of numbers from 0 up to 40 digits before the decimal point, to at most 6 decimal places.
- Decimal values are rounded half up in the last place; the long division method truncates instead.
Privacy
Everything happens in your browser. What you enter or open here is not uploaded or stored by MySmartCoPilot.
Frequently asked questions
How do I simplify a square root?
Find the largest perfect square that divides the number and take its root outside: 72 = 36 × 2, so √72 = √36 × √2 = 6√2. Writing the number as a product of primes and pairing equal factors finds that square for you.
What is the square root of a negative number?
It is not a real number, because every real number squared is zero or positive. In the complex numbers √−1 = i, so √−72 = 6i√2. Cube roots of negative numbers are real: ∛−27 = −3.
How do I know if a number is a perfect square?
Its square root is a whole number — or, for a fraction in lowest terms, its top and bottom are both perfect squares. 144 = 12² is a perfect square; 72 is not, because 8² = 64 and 9² = 81. In prime factors, every exponent of a perfect square is even.
How many cube roots does a number have?
Three, in the complex numbers. 8 has the real cube root 2 and two complex ones, −1 ± i√3 ≈ −1 ± 1.732i. A real number has exactly one real cube root.
How accurate are the decimals?
Every digit shown is correct. Real roots are computed with exact integer arithmetic to one extra place and then rounded, so √2 to 50 places ends …537695, matching published tables of √2.
What does ⁿ√ mean?
It is the nth root: ⁿ√x is the number that gives x when it is multiplied by itself n times. ⁴√81 = 3 because 3 × 3 × 3 × 3 = 81. The square root is the 2nd root and the cube root the 3rd.