Nth Root Calculator
Inputs
| Radicand | 27 |
|---|---|
| Root Index | 3 |
Nth Root Calculator
Calculate the principal nth root of any real number. Supports cube roots, fourth roots, and beyond — with the negative real root shown for even indices.
Inputs
Results
Enter a value to see results.
The nth root of a number x, written ⁿ√x, is the value r that satisfies rⁿ = x. For n = 2 this is the familiar square root; for n = 3 it is the cube root. The calculator computes the principal real nth root and, for even n with positive x, the negative real root as well.
Definition and principal root convention
For a positive radicand x and any integer n ≥ 2, rⁿ = x has exactly two real solutions when n is even (one positive, one negative) and exactly one real solution when n is odd. The principal nth root is defined as the unique real root that satisfies:
- ⁿ√x ≥ 0 when x ≥ 0 (positive or zero result)
- ⁿ√x < 0 when x < 0 and n is odd (extends naturally to negative radicands)
This convention makes ⁿ√x a proper function — one input, one output — rather than a multivalued relation. The radical symbol ⁿ√ always denotes the principal root.
Formula
The principal nth root is computed as:
nx=sign(x)⋅∣x∣1/nwhere sign(x) is +1 for positive x, 0 for zero, and −1 for negative x. For positive x this reduces to x^(1/n). For negative x with odd n, the formula correctly yields a negative result (for example, ∛(−8) = −2).
Nth root as a fractional exponent
The nth root and the exponent 1/n are the same operation:
nx=x1/nThis follows from the power rule: raising x^(1/n) to the nth power recovers x, because (x^(1/n))^n = x^(n/n) = x. The equivalence generalizes: x^(m/n) = ⁿ√(xᵐ), combining a power m and a root n in a single fractional exponent.
Even roots of negative numbers
When n is even, rⁿ is always non-negative for any real r (because an even power of a negative number is positive). Therefore rⁿ = x has no real solution when x < 0. The result is a complex number — a multiple of the imaginary unit i — which lies outside the real number line.
For example, √(−9) = 3i, not −3. The value −3 satisfies (−3)² = 9, not −9, so it is not a square root of −9.
Worked example
Problem: Find the principal 4th root of 81, and identify both real 4th roots.
- Apply the formula: ⁴√81 = 81^(1/4).
- Compute: 3⁴ = 81, so 81^(1/4) = 3.
- The principal 4th root is 3.
- Since n = 4 is even and x = 81 > 0, a second real root exists: −3.
- Verify: (−3)⁴ = 81. ✓
Both 3 and −3 are real 4th roots of 81; the principal root is 3.
Multiple nth roots
Over the real numbers, the count of real nth roots of a positive number x depends on n:
| n | Real nth roots of x > 0 | Real nth root of x < 0 |
|---|---|---|
| odd | 1 (positive) | 1 (negative) |
| even | 2 (positive and negative) | none (complex only) |
Over the complex numbers, every nonzero number has exactly n distinct nth roots, distributed evenly around a circle in the complex plane. The principal root is the one with the smallest non-negative argument (angle).
Relationship to logarithms and exponents
The nth root can also be expressed using logarithms:
nx=e(lnx)/n(x>0)This form is sometimes used in numerical algorithms when a direct power computation is unavailable. For the negative-x odd-n case, compute −(ⁿ√|x|) instead.
Applications
- Geometric scaling: If a cube has volume V, its edge length is ∛V — the cube root of the volume.
- Compound growth: If an investment grows by a factor F over n periods, the per-period factor is ⁿ√F.
- Signal processing: Root-mean-square (RMS) values involve a square root; more general norms use higher-order roots.
- Statistics: The geometric mean of n numbers equals the nth root of their product.
Frequently Asked Questions (FAQ)
Can you take an even root of a negative number?
Not in the real numbers. Even-index roots require the radicand to be non-negative because no real number, when raised to an even power, produces a negative result. For example, both 3² = 9 and (−3)² = 9 are positive, so √(−9) has no real value. The result is a complex number (involving the imaginary unit i), which this calculator does not compute.
What is the difference between √x and the principal nth root?
The square root symbol √ denotes the principal (non-negative) square root, a special case of the nth root with n = 2. The principal nth root is defined as the real nth root that has the same sign as x. For positive x it is always positive; for negative x with odd n it is negative. The convention of "principal" ensures the function returns a single, well-defined value.
How is the nth root related to a fractional exponent?
The nth root and a fractional exponent are the same operation: ⁿ√x = x^(1/n). This equivalence follows from the rule (x^a)^b = x^(ab): (x^(1/n))^n = x^(n/n) = x^1 = x, confirming that x^(1/n) undoes the nth power. More generally, x^(m/n) = ⁿ√(x^m), combining a power and a root in one expression. Scientific calculators use the y^x or x^(1/n) key to compute any root.
Why are there multiple nth roots?
The equation t^n = x can have more than one solution depending on n. For even n and positive x, both t = ⁿ√x and t = −ⁿ√x satisfy t^n = x, so there are two real roots. For odd n there is exactly one real root (either positive or negative, matching the sign of x). Over the complex numbers, every nonzero number has exactly n distinct nth roots, spaced evenly around a circle in the complex plane — but only the real roots matter here.
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