Nth Root Calculator
Nth Root Calculator
The degree must be a positive integer. Negative inputs have real roots only for odd degrees.
Enter your values
Enter values and select Calculate.
How to Use the Nth Root Calculator
Use this free Nth Root Calculator to find the real nth root of a number. Enter the number (x) and the root degree (n), then select Calculate. The root degree must be a positive integer. For negative numbers, a real result is possible only when the degree is odd.
- Enter the number whose root you want to find in the Number (x) field.
- Enter a positive whole number in the Root degree (n) field.
- Select Calculate to display the nth root.
- Select Clear to reset the fields and start another calculation.
- Check whether the result raised to the entered degree gives the original number, allowing for rounding.
What Is an Nth Root?
An nth root is a value that, when raised to the power n, produces the original number. The degree n identifies which root is being calculated. For example, a square root is a second root, and a cube root is a third root. The principal real root is used for positive inputs. For a negative input, an odd degree has a real negative root, while an even degree has no real-number root.
Nth Root Formula
r = ⁿ√x, where rⁿ = x and n is a positive integer.
Here, x is the number, n is the root degree, and r is the root. For real-number calculations, negative x values require an odd value of n.
Worked Examples
Example 1: Fourth root of 81
Number (x) = 81; root degree (n) = 4.
3 × 3 × 3 × 3 = 81, so 3⁴ = 81.
Result: ⁴√81 = 3.
Example 2: Fifth root of −32
Number (x) = −32; root degree (n) = 5.
(−2) × (−2) × (−2) × (−2) × (−2) = −32.
Result: ⁵√(−32) = −2.
Example 3: Sixth root of 64
Number (x) = 64; root degree (n) = 6.
2 × 2 × 2 × 2 × 2 × 2 = 64.
Result: ⁶√64 = 2.
Even Roots and Odd Roots
For a positive number, the principal real nth root is positive. Zero has a real nth root of zero for every positive integer degree. A negative number has a real root when the degree is odd, such as the cube root of −8, which is −2. A negative number does not have a real even root, such as a square root or fourth root, because an even power of a real number cannot be negative.
Where Are Nth Roots Used?
- Mathematics: solving equations that contain powers and radicals.
- Geometry: finding dimensions from area, volume, or higher-dimensional measurements.
- Science: working with mathematical relationships, scaling, and scientific models.
- Finance and statistics: applying root operations in selected growth and statistical calculations.
- Education: checking exercises involving square roots, cube roots, and higher roots.
Frequently Asked Questions
1. What does nth root mean?
The nth root of a number x is a value r such that raising r to the power n gives x.
2. How do I calculate an nth root?
Enter the number and a positive integer root degree, then select Calculate. The calculator returns the real nth root when one exists.
3. What is the difference between square root and nth root?
A square root is a specific type of nth root with degree 2. The nth-root operation can use any positive integer degree.
4. Can I calculate the nth root of a negative number?
Yes, when the root degree is odd. For example, the fifth root of −32 is −2. A negative number has no real root for an even degree.
5. Can the root degree be zero?
No. The root degree must be a positive integer. A zero-degree root is not defined by this calculator's formula.
6. What is the nth root of zero?
The nth root of zero is zero for every positive integer degree.
7. What is the principal root?
The principal real root is the non-negative root selected for a positive number when using an even degree. For odd degrees, the real root has the same sign as the input.
8. How is a cube root related to an nth root?
A cube root is an nth root with degree 3. It finds the number that produces the input when multiplied by itself three times.
9. Can the result be a decimal?
Yes. Many numbers do not have an integer nth root, so the result may be a decimal approximation.
10. Is this Nth Root Calculator free?
Yes. You can use this Nth Root Calculator on SRIKAMI without a fee.
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