Root Calculator finds the nth root of a number for any index n of 2 or greater, covering square roots, cube roots, and every higher-order root on one page. It returns an exact integer when the input is a perfect nth power and a 12-digit decimal approximation otherwise.
Enter a value and an index
Every root problem needs two inputs: the value under the radical and the index of the root. A square root uses index 2, a cube root uses index 3, and so on.
Root Calculator treats the index as its own field rather than hardcoding it, so a single tool answers "what is the seventh root of 128" as easily as "what is the square root of 49."
Find an exact root for a perfect power
When the value is a perfect nth power, the root comes out as a clean integer. The cube root of 125 is 5, since 5^3 = 125 exactly. The sixth root of 64 is 2, since 2^6 = 64 exactly.
Root Calculator checks for this exact match first, using integer search rather than a decimal approximation, so perfect powers never display with unnecessary rounding.
Approximate a root that is not exact
Most values are not perfect nth powers for a chosen index. The cube root of 50 is not a whole number; its exact value is approximately 3.684.
Root Calculator computes this using the fractional exponent form, x^(1/n), and displays the result to 12 significant digits, which is enough precision for virtually any practical use without becoming an unreadable string of digits.
Understand which roots accept negative inputs
Whether a negative value has a real nth root depends entirely on whether n is even or odd. Odd-index roots, such as cube roots and fifth roots, accept negative inputs and return negative results: the cube root of −27 is −3.
Even-index roots, such as square roots and fourth roots, have no real result for a negative input, since raising any real number to an even power always produces a nonnegative value. Root Calculator applies this parity check automatically based on the index entered.
Simplify a root with a perfect-power factor
When a value is not itself a perfect nth power but contains one as a factor, Root Calculator pulls that factor outside the radical.
The cube root of 40 factors as the cube root of (8 × 5), and since 8 = 2^3, this simplifies to 2 times the cube root of 5, a cleaner form than a raw decimal approximation for exact-form work.
Avoid this common mistake
Assuming every root accepts negative inputs, or assuming none do, both lead to errors, since the correct rule depends on whether the index is even or odd. Check the index first: even indices reject negative inputs as undefined; odd indices accept them and return a negative result. Root Calculator enforces this rule directly rather than leaving it to guesswork.
Work through a high-index example
Root Calculator handles indices well beyond the commonly memorized square and cube roots. The seventh root of 128 is exactly 2, since 2^7 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128.
For a non-perfect case, the seventh root of 100 has no whole-number answer, since 2^7 = 128 overshoots and 1^7 = 1 undershoots; the tool reports the approximate value, about 1.931, using the fractional exponent 100^(1/7).
Understand the general nth root formula
Every root, regardless of index, reduces to the same underlying operation: the nth root of x is defined as x^(1/n). A square root is x^(1/2), a cube root is x^(1/3), and a twentieth root is x^(1/20), all computed by the identical exponentiation rule with only the fraction's denominator changing.
This is why a single general-purpose root tool can replace a separate calculator for every specific index, since the math never actually changes, only the number n plugged into the formula.
Frequently asked questions
How do you find the nth root of a number?
To find the nth root of a number, raise it to the power 1/n, or equivalently search for an integer that, raised to the nth power, matches the number exactly for a clean result.
What is the cube root of 125?
The cube root of 125 is 5, since 5 raised to the third power (5 × 5 × 5) equals 125 exactly.
Can every root accept negative numbers?
Not every root accepts negative numbers; only odd-index roots, such as cube roots and fifth roots, accept negative inputs and return a real negative result. Even-index roots, such as square roots and fourth roots, are undefined for negative inputs.
What is the difference between a square root and an nth root?
A square root is the specific case of an nth root where n equals 2, while a general nth root allows any index 2 or greater, covering cube roots, fourth roots, and beyond within the same formula, x^(1/n).
How do you simplify a root that is not a perfect power?
To simplify a root that is not a perfect power outright, factor out the largest perfect nth-power factor from inside the radical and move it outside as a coefficient, leaving a smaller remaining value under the root sign.
What index does a cube root use?
A cube root uses index 3, meaning it asks what number multiplied by itself three times produces the given value. The cube root of 27 is 3, since 3 × 3 × 3 = 27.
What is the seventh root of 128?
The seventh root of 128 is exactly 2, since 2 raised to the seventh power (2 × 2 × 2 × 2 × 2 × 2 × 2) equals 128 exactly.
Is there one formula that covers every root index?
One formula covers every root index: the nth root of x is defined as x raised to the power 1/n, which applies identically whether n is 2, 3, 7, or any other integer of 2 or greater.
Summary
Root Calculator computes the nth root of any real number for an index of 2 or greater, returning an exact integer for perfect powers and a 12-digit decimal approximation otherwise.
It applies the correct rule for negative inputs based on whether the index is even or odd, and it simplifies non-perfect powers by pulling out any perfect-power factor before falling back to a decimal.