Concept · Basic Math
What Is an Nth Root? A Plain Explanation
An nth root asks a multiplication question backward: what, raised to this power, gives this number? The answer changes shape depending on one detail — whether the power is odd or even.
The question a root asks
An nth root asks: what number, raised to the nth power, gives this result? The square root of 9 is 3 because 3² = 9. The cube root of 8 is 2 because 2³ = 8. Every root, whatever the index, is the same question with a different exponent.
- ⁿ√xn is the root index, x is the radicand — the number under the root
- √x means ²√xno index shown is shorthand for a square root, index 2
- ∛x is the cube root, index 3and so on for any whole number index
Why even and odd roots behave differently
Squaring a negative number gives a positive result — (−3)² = 9, the same as 3². This means a positive number under a square root has two numbers that square to it, one positive and one negative, and by convention √x returns only the positive one. It also means a negative number has no real square root at all, since no real number squared gives a negative result.
Cubing a negative number keeps it negative — (−2)³ = −8, not 8. This breaks the symmetry that caused the problem with square roots: every real number, positive or negative, has exactly one real cube root. The same holds for any odd index; the same restriction as square roots holds for any even index.
| Index | Root of a negative number |
|---|---|
| even (2, 4, 6, …) | not a real number |
| odd (3, 5, 7, …) | real, and negative |
When a root simplifies, and when it doesn't
24's cube root simplifies because 24 = 8 · 3, and 8 is a perfect cube (2³). Pulling the perfect cube out leaves ∛24 = 2∛3 — a whole number times a smaller root. Ten's cube root does not simplify this way, because no factor of 10 is a perfect cube; it stays as an irrational number, roughly 2.154, that cannot be written exactly as a fraction or a finite decimal.
Perfect powers, the building blocks
A perfect square is what you get by squaring a whole number — 1, 4, 9, 16, 25. A perfect cube comes from cubing one — 1, 8, 27, 64, 125. Every index has its own list, and recognising a few of the small ones by sight is what makes spotting a simplification fast instead of a search.
| Index | First few perfect powers |
|---|---|
| 2 (square) | 1, 4, 9, 16, 25, 36, 49, 64 |
| 3 (cube) | 1, 8, 27, 64, 125, 216 |
| 4 | 1, 16, 81, 256, 625 |
How a root and a fractional exponent are the same thing
ⁿ√x can always be written as x^(1/n) — the two notations mean exactly the same number. This is not a coincidence of notation; it follows from the rule that exponents add when you multiply same-base powers: (x^(1/n))ⁿ = x^(n/n) = x¹ = x, which is precisely what an nth root is defined to satisfy.
This equivalence is why a calculator with only an exponent key, no root key, can still compute any root — enter the radicand raised to 1 divided by the index.
Questions about what is an nth root
What does the small number in front of a radical mean?
The root index — which root is being taken. No number shown means a square root by convention; a small 3 means a cube root, and so on for any whole number.
Why does a negative number have no square root but does have a cube root?
Squaring always gives a non-negative result, so no real number squares to a negative — a square root of a negative number does not exist among the reals. Cubing preserves the sign, so every real number, negative included, has exactly one real cube root.
Is √9 equal to both 3 and −3?
By definition, no — √9 means the positive square root, 3, by convention. It is true that both 3 and −3 square to 9, which is why the equation x² = 9 has two solutions, ±3, even though √9 itself names only one of them.
Is an nth root the same as a fractional exponent?
Yes. ⁿ√x and x^(1/n) are the same number, always. This equivalence is why calculators without a dedicated root key can compute any root using exponentiation instead.