Nth Root Guide — Square, Cube, and Higher Roots Explained

What is an nth root?

The nth root of x is the number r that satisfies rⁿ = x. For n=2 it's the square root; for n=3 the cube root; for any n, the nth root.

The square root of 9 is 3 because 3² = 9. The cube root of 8 is 2 because 2³ = 8. The fourth root of 81 is 3 because 3⁴ = 81.

In general, ⁿ√x = r means rⁿ = x. The index n is what distinguishes the operation: 2 = square, 3 = cube, 4 = fourth, and so on.

Most calculators (including this one) return the principal root — the non-negative value for even indices. Technically, x² = 4 has two solutions (x = 2 and x = −2), but √4 by convention means just 2.

Perfect roots — when the answer is exact

If x = kⁿ for some integer k, then ⁿ√x = k exactly — no decimals, no approximation.

Perfect squares are the easiest to spot: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … Each is the square of an integer, so its square root is that integer.

Perfect cubes grow faster: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, …

Perfect fourth powers grow faster still: 1, 16, 81, 256, 625, 1296, 2401, …

The calculator detects these automatically — if the input matches a perfect nth power, you get the exact integer instead of a decimal approximation.

Simplifying radicals

A radical can often be pulled apart into an integer factor outside the root and a smaller radicand inside.

The trick: factor the radicand and look for perfect nth-power factors. If x = kⁿ · m, then ⁿ√x = k · ⁿ√m.

For square roots, look for perfect-square factors (4, 9, 16, 25, …). For cube roots, look for perfect-cube factors (8, 27, 64, …).

This matters for algebra: "simplify √72" isn't the same question as "what's √72 as a decimal?" The exact answer 6√2 preserves precision; the decimal 8.485… loses it.

Negative radicands — odd vs even index

Odd-index roots of negatives are real. Even-index roots of negatives are complex — the calculator returns the principal complex root automatically.

Odd powers preserve sign: (−2)³ = −8, so ³√−8 = −2. Works for any odd index: ⁵√−32 = −2.

Even powers are always non-negative: (−2)² = 4, not −4. So there's no real number whose square is −4 — but there is a complex one: √−4 = 2i (and −2i).

The calculator returns the principal complex root for even-index negative radicands. For n=2 it's purely imaginary (√−4 = 2i); for higher even n like 4, the principal root has both real and imaginary parts (⁴√−16 = √2 + √2·i). The general formula is |x|^(1/n) · (cos(π/n) + i·sin(π/n)).

Four common mistakes

Root mistakes tend to cluster around a few specific misreads of the algebra rules.

1. Distributing roots over addition. √(a + b) ≠ √a + √b. Try it: √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. Only products distribute: √(a·b) = √a · √b.

2. Forgetting the ± in equations. √x and x² are not inverse for all x. In x² = 4, x is ±2, not just 2. The √ symbol returns only the principal (non-negative) root by convention.

3. Mixing simplified and decimal forms. 5√2 and 7.071 refer to the same number — but one preserves exactness. In algebra, prefer the simplified radical; in applied contexts, the decimal.

4. Confusing radicand with index. In ⁿ√x, n is the index (the small number above the radical symbol) and x is the radicand (the number inside). Don't swap them: ³√8 = 2 but ⁸√3 ≈ 1.147.