5th Root Calculator
How to Use the Fifth Root Calculator
- Enter the number (x) you want to find the fifth root of.
- Click the “Calculate” button.
- View the fifth root instantly.
Features of the Fifth Root Calculator
- Instantly calculates the fifth root of any real number.
- Works with positive numbers, negative numbers, and decimals.
- Correctly handles negative inputs, since fifth roots (like all odd roots) are defined for negative numbers.
- Free to use with no sign-up required.
Fifth Root Formula
The fifth root of a number x is the value that, when multiplied by itself five times, equals x.
Fifth Root = ⁵√x
Since 5 is an odd number, fifth roots are defined for negative numbers too, unlike square roots or fourth roots, since a negative number multiplied by itself an odd number of times stays negative.
Common Fifth Roots (Reference Table)
| Number (x) | Fifth Root (⁵√x) |
|---|---|
| 1 | 1 |
| 32 | 2 |
| 243 | 3 |
| 1024 | 4 |
| 3125 | 5 |
| 100000 | 10 |
| 0.5 | 0.870551 |
| 2 | 1.148698 |
| −32 | −2 |
| −243 | −3 |
How to Calculate a Fifth Root by Hand
To find the fifth root of a perfect fifth power, identify which whole number multiplied by itself five times gives the original number. For numbers that aren’t perfect fifth powers, the fifth root is found using estimation, a calculator, or the exponent 1/5.
Suppose you want the fifth root of 243. Since 3 × 3 × 3 × 3 × 3 = 243, the fifth root of 243 is 3.
Thus, ⁵√243 = 3.
Solved Examples on Fifth Roots
1. Fifth Root of a Perfect Fifth Power
Example: Find the fifth root of 1024.
Since 4 × 4 × 4 × 4 × 4 = 1024, ⁵√1024 = 4.
2. Fifth Root of a Negative Number
Example: Find the fifth root of −243.
Since (−3) × (−3) × (−3) × (−3) × (−3) = −243, ⁵√−243 = −3.
3. Fifth Root of a Non-Perfect Fifth Power
Example: Find the fifth root of 10.
10 is not a perfect fifth power, so ⁵√10 ≈ 1.584893.
4. Fifth Root of a Decimal
Example: Find the fifth root of 0.5.
⁵√0.5 ≈ 0.870551.
1. What is a fifth root?
A fifth root of a number x is a value that, when multiplied by itself five times, produces x. It is written as ⁵√x or x^(1/5).
2. Can negative numbers have fifth roots?
Yes. Since 5 is an odd number, fifth roots are defined for negative numbers, because a negative number raised to an odd power stays negative. For example, ⁵√−32 = −2.
3. How is a fifth root different from a square root?
A square root asks what number multiplied by itself twice gives x, while a fifth root asks what number multiplied by itself five times gives x. Square roots of negative numbers aren’t real, but fifth roots of negative numbers are, since 5 is odd and 2 is even.
4. What is a perfect fifth power?
A perfect fifth power is a number that results from multiplying an integer by itself five times, such as 1, 32, 243, 1024, and 3125. Their fifth roots are always whole numbers.
5. Does the calculator work with decimals?
Yes, the calculator accepts decimal values and returns results rounded to eight decimal places for precision.
6. Where are fifth roots used in practice?
Fifth roots appear in solving quintic-related equations, in engineering scaling calculations involving fifth-power relationships, and in advanced algebra when simplifying radical expressions.