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Square Root & N-th Root calculator →

Square Root & N-th Root guide

The Root Calculator finds square roots, cube roots, and any N-th root. It provides both the decimal approximation and the simplified radical form (for square roots).

Worked examples

Simplify the square root of 50

  1. Set the Root Index (n) to 2.
  2. Enter 50 as the Number.
  3. Click Calculate.
Result: √50 = 5√2 ≈ 7.071068

The calculator finds the largest perfect square factor of 50, which is 25. The square root of 25 is 5, leaving √2 inside the radical.

Calculate the cube root of a negative number

  1. Set the Root Index (n) to 3.
  2. Enter -8 as the Number.
  3. Click Calculate.
Result: ∛-8 = -2

Unlike square roots, odd roots of negative numbers are valid real numbers. -2 × -2 × -2 = -8.

Find the 4th root of 16

  1. Enter 4 as the Root Index.
  2. Enter 16 as the Number.
  3. Click Calculate.
Result: ⁴√16 = 2

The 4th root asks: what number multiplied by itself 4 times equals 16? The answer is 2.

Tips

  • •Use the Cheat Sheet button to quickly view a table of square roots from 1 to 100.
  • •The simplified radical form is only available for standard square roots (n=2).
  • •You cannot take an even root of a negative number in standard real math.

Frequently asked questions

What is a square root?
A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 × 3 = 9.
What is the difference between a square root and an N-th root?
A square root (index 2) asks what number multiplies by itself to get the target. An N-th root asks what number multiplies by itself N times.
Can you take the square root of a negative number?
In standard real math, no. The square root of a negative number results in an imaginary number (using the symbol i). The calculator rejects negative inputs for even roots.
Why can I take the cube root of a negative number?
Because multiplying a negative number by itself three times results in a negative number (-2 × -2 × -2 = -8). Odd roots work with negatives.
What is the simplified radical form?
It rewrites a square root by pulling out any perfect squares. For example, √50 becomes 5√2. This preserves the exact mathematical value instead of rounding.
Is the decimal result rounded?
Yes, the decimal is rounded to 6 decimal places for readability. The radical form (if available) is exact.
What is a perfect square?
A number whose square root is a whole integer. Examples include 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
What does "undefined" mean in this context?
It means the operation cannot be performed in real numbers. For example, taking an even root of a negative number is undefined in real math.
What is the root index?
The root index (n) tells the calculator what type of root to take. Index 2 is a square root, 3 is a cube root, 4 is a fourth root, etc.
Can the root index be a decimal?
Yes. A root index of 2.5 is mathematically equivalent to raising the number to the power of 1/2.5 (or 0.4). The calculator handles this.
What is the principal root?
When you take the square root of a positive number, there are technically two answers (positive and negative). The principal root is the positive one.
How do I calculate the square root of a fraction?
You can take the square root of the numerator and the denominator separately. For example, √(9/16) = 3/4.
Why does √2 have infinite decimal places?
It is an irrational number. It cannot be expressed as a simple fraction, so its decimal representation never ends and never repeats.
Can I copy the radical or the decimal?
Yes, there are separate Copy buttons for both the decimal approximation and the exact radical form.
How large can the number be?
You can enter very large numbers, but extremely large inputs may lose decimal precision due to JavaScript floating-point limits.
What is the 0th root?
The 0th root is mathematically undefined because you cannot find a number that, when raised to the power of 0, equals anything other than 1.
Does this tool support complex numbers?
No. This calculator only handles real numbers. For complex numbers, use the dedicated Complex Numbers calculator.
Can I use the keyboard to enter data?
Yes, you can type the numbers and press Enter to run the calculation.
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