Nth Root Calculator
Every root is a fractional power.
Work out Nth Root. Every root is a fractional power. Shows the working, not just the answer.
2 for a square root, 3 for a cube root
Root
5.19615242
Not a whole number; the nearest is 5
Any root can be written as a fractional power: the nth root of x is x to the power 1/n. That identity is what lets a calculator handle roots that are not whole numbers at all. Note that 5.1962 is the principal root — -5.1962 squared gives the same answer, and the radical sign conventionally means only the positive one.
How the Nth Root Calculator works
Any root of any number, from square roots upwards, with an exactness check. Every root is a fractional power — the nth root of x is x to the power 1/n — which is how roots that are not whole numbers are handled at all.
Also known as: fourth root calculator · fifth root calculator · radical calculator · how to calculate any root
Every root is a fractional power
The nth root of x is x to the power 1/n. That single identity collapses roots and powers into one operation, which is why a calculator needs only a power key and why fractional exponents work at all.
It extends further. The nth root of x to the power m is x to the power m/n, so the cube root of x squared and the square of the cube root of x are the same thing — which is not obvious until the exponents are written out.
Negative exponents invert: x to the power −1/2 is one over the square root of x. The whole system is consistent, and it is what lets a calculator handle roots that are not whole numbers at all.
Simplifying radicals and why surds are left alone
The root of a product is the product of the roots, for non-negative values. That identity is what simplifying depends on: the square root of 72 is the root of 36 times 2, which is 6 times the root of 2.
The root of a sum is not the sum of the roots, and confusing the two is one of the most common algebra errors. The root of 9 plus the root of 16 is 7; the root of 25 is 5. They are not the same operation.
A surd is a root that cannot be written exactly as a fraction — the square root of 2 is the classic case, proved irrational by the Pythagoreans. Exact working leaves surds in radical form precisely because any decimal is an approximation.
How roots were computed before calculators
By logarithms. Taking the log of a number, dividing by the root index, and taking the antilog gives the root — turning an awkward operation into a division, which is what log tables were for.
That is also what a slide rule does mechanically. The scales are logarithmic, so sliding them adds and subtracts logs, which multiplies and divides the underlying numbers.
There were also digit-by-digit algorithms resembling long division, taught for square roots into the twentieth century. They work and they are slow, and calculators removed the reason to know them within a single generation.
Where to go next
The Nth Root question rarely arrives on its own. These are the ones that usually come with it:
- Cube Root Calculator — Odd roots of negatives are real; even roots are not.
- Quadratic Formula Calculator — The discriminant decides everything before you solve.
- Rounding Calculator — Half-up and half-even disagree at exactly the half.
- Percentage Calculator — Every common percentage question in one place.
Frequently asked questions
How do I calculate the nth root?
Raise the number to the power of 1/n. The fourth root of 16 is 16 to the power 0.25, which is 2. Any calculator with a power key does roots.
Can the root itself be a fraction?
Yes. A root of 0.5 is the same as raising to the power of 2, so the 'half root' of 9 is 81. It is unusual notation and it is arithmetically consistent.
Why does an even root have two answers?
Because both a positive and a negative number square to the same positive value. The radical sign conventionally means only the positive one, called the principal root — which is why an equation asks for ± explicitly.
What is a surd?
A root that cannot be written exactly as a fraction — the square root of 2 is the classic example. Surds are left in radical form in exact working precisely because any decimal is an approximation.
How do I simplify a radical?
Factor out the largest perfect power. The square root of 72 is the square root of 36 times 2, which is 6 times the square root of 2.
Is the nth root the same as dividing by n?
No, and it is a common confusion. Dividing 64 by 3 gives 21.3; the cube root of 64 is 4. Roots undo powers, they do not undo multiplication.
What is a principal root?
The non-negative root of a positive number, which is what the radical sign conventionally means. It is why the square root of 9 is written as 3 rather than ±3, and why equations state the ± explicitly.
How do roots and powers interact?
The nth root of x to the power m is x to the power m/n. It means fractional exponents handle every root and power in one notation, which is why calculators only need a power key.
Can a root be negative?
The index can be, which inverts as well as roots: x to the power −1/2 is one over the square root of x. The result of an even root of a positive is conventionally taken as positive.
Why can't I average roots?
Because roots are not linear. The square root of 9 plus the square root of 16 is 7, while the square root of 25 is 5 — a common and consequential mistake in algebra.
What is the root of a product?
The product of the roots: √(ab) = √a × √b for non-negative values. That identity is what simplifying radicals depends on.
How were roots calculated before calculators?
By tables of logarithms, which turn a root into a division. Taking the log, dividing by the index and taking the antilog was the standard method for centuries.
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