Use this root calculator to find square, cube, or n-th roots, see an approximate decimal, and check whether a real answer exists.
How to use our Root Calculator
- Enter the value in Number under the root (radicand).
- Enter the whole-number root in Root index (n). Use 2 for square root, 3 for cube root, 5 for fifth root, and so on.
- Choose Decimal places to control how many digits appear in Approximate root.
- Click Calculate to see Approximate root, Exact simplified form, Check by raising answer to n, and the Real-number note.
- If Real-number note says the root is not real, read the result as N/A for this calculator instead of expecting a decimal answer.
- Use Exact simplified form when you want a math-class answer like 6sqrt(2), 1/2, or 2 instead of only a rounded decimal.
- Sanity-check the result by comparing Check by raising answer to n to your original radicand. It should match exactly for perfect roots or be very close when the decimal answer is rounded.
Definitions
Number under the root (radicand): The number you are taking a root of, such as 72 in sqrt(72).
Root index (n): The kind of root you want. For example, 2 means square root, 3 means cube root, and n means n-th root [2].
Approximate root: The decimal version of the answer, rounded to the number of digits you chose.
Exact simplified form: An answer written exactly, such as 2, 1/2, or a simplified radical like 6sqrt(2), instead of a rounded decimal.
Check by raising answer to n: A quick test where the displayed decimal root is multiplied by itself n times to see if it returns close to the original radicand.
Real-number note: A plain-language message that explains whether the requested root exists in the real numbers. Odd roots of negative numbers are real, but even roots of negative numbers are not real [1].
Common mistakes and quick fixes
Mistake: Typing a decimal like 2.5 in Root index (n) .
Fix: Enter a whole number greater than 1 in Root index (n) , such as 2, 3, 4, or 5.
Mistake: Expecting a real value in Approximate root for a negative radicand with an even Root index (n) , such as -16 with 2.
Fix: Read the Real-number note . In real numbers, even roots of negative values are not real, so this calculator correctly shows N/A.
Mistake: Thinking Check by raising answer to n must match the radicand perfectly every time.
Fix: If Approximate root is rounded, the check can be slightly off. Increase Decimal places if you want a closer match.
Mistake: Ignoring Exact simplified form when the answer is a simple exact value like 2 or 1/2.
Fix: Use Exact simplified form for classwork and exact notation, and use Approximate root when you need a decimal.
Mistake: Entering the wrong value in Number under the root (radicand) , such as typing 36 when the problem is sqrt(72).
Fix: Put the full number under the radical into Number under the root (radicand) , then let Exact simplified form show any simplification.
Mistake: Using a blank or non-numeric entry in Number under the root (radicand) or Decimal places .
Fix: Enter a finite number for Number under the root (radicand) and a whole number from 0 to 15 for Decimal places .
Limitations & Key Assumptions / Boundary Conditions
- This calculator works in the real-number system only. If Number under the root (radicand) is negative and Root index (n) is even, it shows no real answer instead of complex roots.
- Root index (n) must be a whole number greater than 1. Values like 1, 0, negative numbers, or decimals are treated as input errors.
- Approximate root is rounded to the chosen Decimal places, so Check by raising answer to n may be close to the radicand rather than exactly equal.
- Exact simplified form is shown when a simple exact real form is practical, such as an integer, a fraction, or a simplified radical. Some numbers do not simplify neatly, so the calculator may mainly rely on the decimal result.
- Very large or very small inputs can be harder to display neatly because decimal expansions may be long or highly rounded.
Methodology
How the calculator finds the root
This calculator uses the standard idea that an n-th root of x is a number r such that r multiplied by itself n times equals x [1].
r = x^(1/n)
r^n = x
Here, x is the radicand, n is the root index, and r is the root result.
Real-number domain check
Before calculating, the tool checks whether a real answer is allowed.
If n is even and x < 0, no real n-th root exists.
If n is odd, r = sign(x) * |x|^(1/n)
That is why the cube root of -8 is -2, but the square root of -16 has no real result in this calculator [2].
Approximate root and verification
When a real root exists, the calculator computes a decimal answer and rounds it to the chosen number of decimal places.
check = r_approx^n
The check uses the rounded decimal value, so it may be very close to the original radicand instead of perfectly exact.
Exact simplified form
When possible, the calculator also looks for a simpler exact form by pulling out perfect n-th power factors.
nth_root(a*b) = nth_root(a) * nth_root(b) when a = k^n
Example:
sqrt(72) = sqrt(36*2) = 6sqrt(2). Another example:
the fifth root of 32 is 2 because 2^5 = 32.
If the radicand is a simple fraction in decimal form, the calculator may also show an exact fraction. For example, sqrt(0.25) = 1/2.
Worked mini-example
Suppose Number under the root (radicand) is 72, Root index (n) is 2, and Decimal places is 6.
r = 72^(1/2) = sqrt(72) = 8.485281...
rounded r = 8.485281
check = 8.485281^2 = 71.999993...
exact simplified form = 6sqrt(2)
So the decimal answer is about 8.485281, the check is very close to 72, and the exact simplified form is 6sqrt(2).
Input rules used
The calculator accepts only finite real numbers for the radicand, requires Root index (n) to be an integer greater than 1, and requires Decimal places to be a whole number from 0 to 15.