Radicals Calculator (Nth Root)

What a radical is

A radical is the root symbol $\sqrt[n]{x}$, read as "the $n$th root of $x$." The number $x$ under the bar is the radicand; the small number $n$ tucked into the notch is the index, telling you which root to take:

$\sqrt[n]{x} = y \quad \text{means} \quad y^n = x$

When $n = 2$, convention drops the index and the symbol is written simply $\sqrt{x}$ — a square root. When $n = 3$ it's a cube root, $\sqrt[3]{x}$. Every other whole-number index from $4$ upward is written explicitly, e.g. $\sqrt[5]{x}$, $\sqrt[6]{x}$, and so on. This calculator generalizes both special cases (and every index up to $20$) into a single tool: pick any radicand and any index, and it returns the exact simplified radical plus a decimal approximation.

The even/odd sign rule

Whether a negative radicand produces a real or an imaginary answer depends entirely on whether the index is even or odd:

Even index, negative radicand → imaginary. Raising any real number to an even power always produces a non-negative result — a negative number times itself an even number of times cancels its own sign in pairs. So no real number, raised to an even power, can ever land on a negative $x$ — there is simply no real $n$th root to find. The answer is instead written with the imaginary unit $i = \sqrt{-1}$: $\sqrt[4]{-16} = \sqrt[4]{16} \times i = 2i$.

Odd index, negative radicand → real. An odd power preserves the sign of its base — a negative number raised to an odd power stays negative, e.g. $(-2)^5 = -32$. So every real radicand, positive or negative, has exactly one real $n$th root when the index is odd, and the sign simply folds into the answer: $\sqrt[5]{-32} = -2$, because $(-2)^5 = -32$ exactly.

Simplifying a radical, step by step

A radical is fully simplified once every prime factor left under the root appears fewer than $n$ times — any complete group of $n$ identical factors has already been pulled out as a whole-number coefficient. To simplify $\sqrt[4]{48}$ by hand:

1. Factor into primes: $48 = 2^4 \times 3$.
2. Group into sets of the index (here, 4): the $2^4$ is one complete group of four 2s; the single $3$ has no group to join.
3. Pull each complete group out, taking its $n$th root: $2^4 \to 2$, becoming the coefficient.
4. Leave the leftover factor under the root: the unpaired $3$ stays inside, giving $\sqrt[4]{48} = 2\sqrt[4]{3}$.

This calculator shows the same prime-factorization breakdown for whatever radicand and index you enter — including negative radicands, where the sign either stays inside the coefficient (odd index) or attaches an "$i$" after simplifying the positive part (even index).

Perfect nth powers

A radicand is a perfect $n$th power when the radical reduces all the way to a whole number with nothing left under the root — for example $\sqrt[5]{32} = 2$ exactly, since $2^5 = 32$. The table below shows a few perfect powers for different indices:

Most radicands are not perfect $n$th powers, so the exact answer stays partially or fully under the root as a simplified radical, with a decimal approximation alongside it.

Choosing an index

The index picker on this calculator covers every whole-number index from $2$ to $20$ — far more than the square and cube roots covered by the dedicated Square Root and Cube Root calculators. Higher indices show up less often day-to-day, but they follow exactly the same rules: pick the index, and the even/odd sign rule and the simplification steps above apply unchanged.

Radicals and exponents

A radical is the same operation as a fractional exponent: $\sqrt[n]{x} = x^{1/n}$. Taking the 5th root of $32$ is identical to raising $32$ to the power $\frac{1}{5}$ — both ask "what number, multiplied by itself 5 times, gives $32$?" This connection is why radicals and exponents are inverse operations of each other, and why the rules for combining exponents (like $x^{1/n} \times x^{1/n} = x^{2/n}$) carry over directly to radical notation.

Worked examples

Example 1 — perfect 5th power: $\sqrt[5]{32} = 2$, exactly, since $2^5 = 32$.

Example 2 — simplified radical: $\sqrt[4]{48} = 2\sqrt[4]{3} \approx 2.632148$, since $48 = 2^4 \times 3$.

Example 3 — odd index, negative radicand (real): $\sqrt[5]{-32} = -2$, a real result, since $(-2)^5 = -32$.

Example 4 — even index, negative radicand (imaginary): $\sqrt[4]{-16} = 2i$, an imaginary result, since no real number raised to the 4th power can be negative.

Where higher-index radicals show up

Beyond the everyday square and cube roots, higher-index radicals appear in compound-interest math (finding a per-period growth rate from a total return over $n$ periods is literally an $n$th root), engineering formulas involving higher-order scaling, and algebra problems that ask you to undo an exponent like $x^7 = 128$ by taking $\sqrt[7]{128}$. Whatever the index, the Square Root and Cube Root calculators handle the two most common cases with the same simplification logic used here.