Root Calculator
Calculate square roots, cube roots, and other nth roots.
Understanding Root Calculator
This Root Calculator finds the square root, cube root, or any other nth root of a number. Enter the number you want to calculate and the root degree, then select Calculate to get the result with the mathematical steps and verification.
A root is the inverse operation of raising a number to a power. The nth root of a number x is the value that, when multiplied by itself n times, produces x. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27.
Using this calculator with your own numbers
Root Calculator is intended to show the mathematical relationship behind the answer, not just return a number. Enter the quantities requested by the selected method—such as Number, Root Degree—and keep their units or mathematical meaning consistent. Where this calculator offers different shapes, methods, or forms, choose the one that matches the problem before entering the values.
Why the inputs matter
For a useful check, look at the units and the scale of the answer before accepting it. A formula can be evaluated correctly while an input is still inappropriate for the problem. Repeating the calculation with a simple test value or checking the result against the expected direction of change can help identify an input mistake.
Root Calculator
This Root Calculator finds the square root, cube root, or any other nth root of a number. Enter the number you want to calculate and the root degree, then select Calculate to get the result with the mathematical steps and verification.
A root is the inverse operation of raising a number to a power. The nth root of a number x is the value that, when multiplied by itself n times, produces x. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27.
What Is an nth Root?
The nth root of a number can be written using the radical symbol as ⁿ√x. The small number n is called the index or degree of the root, while x is the radicand. The result is the number that produces the radicand when raised to the nth power.
In equivalent power notation, taking an nth root is the same as raising the number to the power of 1/n. This relationship makes roots closely connected to fractional exponents.
Square Roots
A square root has a degree of 2. It asks which number, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5² = 25.
Square roots are commonly used in geometry, algebra, statistics, physics, engineering, and many other mathematical applications.
Cube Roots
A cube root has a degree of 3. It identifies the number that must be multiplied by itself three times to produce the given number. For example, the cube root of 27 is 3 because 3³ = 27.
Unlike even roots, cube roots and other odd roots can produce real results for negative numbers. For example, ∛(-8) = -2 because (-2)³ = -8.
General nth Roots
The same principle applies to roots of any positive whole-number degree. A fourth root asks which number raised to the fourth power produces the given value, while a fifth root asks which number raised to the fifth power produces it.
yn = x
For example, the fourth root of 81 is 3 because 3⁴ = 3 × 3 × 3 × 3 = 81.
Roots and Exponents
Roots can be expressed as fractional exponents. The denominator of the fractional exponent represents the root degree.
x1/3 = ∛x
x1/n = ⁿ√x
More generally, a fractional exponent can combine a root and a power. For example, xm/n represents the nth root of x raised to the mth power, when the expression is defined in the applicable number system.
Negative Numbers and Roots
Whether a negative number has a real root depends on the degree of the root. Odd roots of negative numbers are real because an odd power preserves the negative sign.
(-2)³ = -8
Even roots of negative numbers do not have real-number solutions. For example, there is no real number whose square is -9. Such results require complex numbers rather than real numbers.
How the Root Calculator Works
The calculator converts the nth-root operation into an equivalent fractional exponent using the relationship ⁿ√x = x1/n. It then evaluates the expression numerically and displays the result with decimal precision.
The calculated result is also raised back to the selected root degree as a verification step. This provides a useful check that the calculated value reproduces the original number, apart from small differences caused by decimal rounding.
Examples
Square root: For 49 with a root degree of 2, √49 = 7 because 7² = 49.
Cube root: For 125 with a root degree of 3, ∛125 = 5 because 5³ = 125.
Fourth root: For 81 with a root degree of 4, ⁿ√81 = 3 because 3⁴ = 81.
Negative cube root: For -27 with a root degree of 3, ∛(-27) = -3 because (-3)³ = -27.
Common Root Values
√9 = 3
√16 = 4
√25 = 5
∛8 = 2
∛27 = 3
∛64 = 4
ⁿ√81 = 3
Root Calculator Formula
The primary formula used by this calculator is:
where n is the root degree. The equivalent defining relationship is:
For real-number calculations, an even root of a negative number is not defined as a real value, while odd roots can return real negative results.
Disclaimer
This calculator is intended for informational purposes only. While we strive for accuracy, we cannot guarantee that every result is correct for every situation.