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Square Root & N-th Root Calculator

Find square roots and radical values.

Root Inputs

Configure root index and target number.

Result
Decimal
—
Simplified Radical Form
Not applicable

Calculation Breakdown:

Enter values to compute result.

About this topic

Square Root & N-th Root Calculate the root of any real number.

A root asks which number, raised to a chosen power, produces the original value. The square root asks for a number that multiplies by itself to make the radicand; for example, √9 = 3 because 3 × 3 = 9. The cube root asks for a number multiplied by itself three times, so ∛(−8) = −2. More generally, the N-th root uses the index N, and the parity of that index determines whether a negative real number has a real root.

Key formulas & rules

  • •Square root = n=2
  • •Cube root = n=3
  • •Negative root if base < 0 and n is odd
  • •Undefined for negative base and even n
  • •Decimal approximation provided

Examples

Input Output Note
√9 3 Square root
∛-8 -2 Negative cube root
∜16 2 Fourth root
√2 1.4142 Irrational

Real-world uses

Pythagorean theorem Standard deviation Engineering
Simplify Radical Form Pull perfect squares out of a square root.

Simplifying a radical means rewriting a root in a form that is easier to work with while keeping its exact value. Look for the largest perfect-square factor inside a square root, take its square root, and leave the remaining factor under the radical. For example, √50 = √(25 × 2) = 5√2. This is often preferable to a rounded decimal because it preserves precision and makes later algebraic calculations easier.

Key formulas & rules

  • •Only applies to square roots (n=2)
  • •Finds largest perfect square factor
  • •Pulls it outside the radical
  • •Preserves exact value
  • •Leaves remainder inside

Examples

Input Output Note
√50 5√2 25 × 2
√72 6√2 36 × 2
√9 3 Perfect square
√7 √7 Cannot simplify

Real-world uses

Exact geometry Trigonometry Surds

Modes & formulas

1

N-th Root

• Calculates the root of any real number. • Enter the number (radicand) and the root index (n). • The calculator uses x^(1/n) to find the root.
ⁿ√x = x^(1/n)

For a square root, n=2. For a cube root, n=3. If the number is negative and the root index is odd, the result will be negative (e.g., ∛(−8) = −2).

2

Simplify Radical

• Simplifies square roots into their most basic radical form. • The calculator finds perfect-square factors within the number. • It pulls the perfect squares outside the radical symbol.
√50 = 5√2

This preserves exact math, avoiding decimal approximations. The tool extracts the largest perfect square (e.g., 25 from 50) and leaves the remainder inside the root.

How to use it

Enter the number and the root index. The tool provides both the decimal approximation and the simplified radical form (if applicable).

Frequently asked questions

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 'i').
Why is the radical form useful?
It preserves exact values. √2 is an irrational decimal, but leaving it as √2 keeps the precision intact.
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.
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.

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Read the full Square Root & N-th Root guide — examples & 18 FAQs