Root Calculator

What this calculator does

The Root Calculator finds a square root, a cube root, or a general nth root of a number. A root asks which value, multiplied by itself a given number of times, produces the original number. The square root asks for a value multiplied by itself twice, the cube root asks for a value multiplied by itself three times, and the general root lets you choose the number of repetitions.

Roots appear in geometry, statistics, physics, finance, and computer graphics. For example, the side length of a square is the square root of its area, and the edge length of a cube is the cube root of its volume.

How to use this calculator

1. Choose Square root, Cube root, or General root.
2. Enter the number. For a general root, also enter the root index, such as 4 for a fourth root.
3. Click Calculate.
4. Read the root in the result panel.

The default square-root example is √9 = 3. The cube-root example is ∛8 = 2, and the general-root example is ³√27 = 3.

Root formulas

Square root:  √x = x^(1/2)
Cube root:    ∛x = x^(1/3)
General root: ⁿ√x = x^(1/n)

For a negative number, an odd root keeps the negative sign. For example, ∛(−8) = −2. An even root of a negative number is not a real number, so the calculator reports an error instead of returning a complex result.

What the results mean

The result is the value whose nth power equals the number you entered. If the result is a whole number, the original number is a perfect power. For example, 9 is a perfect square and 27 is a perfect cube. Otherwise the result is irrational in most cases and is displayed as a rounded decimal.

The calculator displays up to ten decimal places. It does not round intermediate steps, so the displayed value is the closest practical representation of the exact root.

Assumptions and limitations

• Only real roots are calculated. Complex roots are outside the scope of this page.
• The root index must be a whole number from 2 to 1,000.
• The number may be positive, negative, or zero, but an even root of a negative number is rejected.
• Very large or very small values may lose precision because JavaScript uses double-precision floating-point numbers.
• The calculator does not simplify radicals into exact symbolic form. For an exact symbolic answer, use a computer algebra system.

Frequently asked questions

What is the difference between a square root and a cube root?

A square root finds a value that produces the original number when multiplied by itself twice. A cube root finds a value that produces the original number when multiplied by itself three times. The cube root of a negative number is real, while its square root is not.

Can I calculate a fourth root or a tenth root?

Yes. Choose General root and enter the index you need. For a fourth root, enter 4; for a tenth root, enter 10.

Why do I get a decimal instead of a whole number?

Most roots are irrational. A whole-number result appears only when the original number is a perfect power for the chosen index. The decimal is an approximation rounded for display.

Why is the even root of a negative number rejected?

No real number squared, raised to the fourth power, or raised to any other even power can be negative. The calculator stays within real numbers and asks you to use an odd index or a positive number instead.

Source

Reference: Calculator.net — Root Calculator.