Simplify Radicals Calculator

What "simplify $y \cdot \sqrt[n]{x}$" means

The expression $y \cdot \sqrt[n]{x}$ pairs an outer coefficient $y$ with a radical: $\sqrt[n]{x}$, read "the $n$th root of $x$," where $x$ is the radicand and $n$ is the index. Simplifying it means rewriting the radical part in its smallest possible form — pulling every complete group of $n$ matching prime factors out of $x$ — and then combining whatever comes out with the existing coefficient $y$:

$y \cdot \sqrt[n]{x} = y' \cdot \sqrt[n]{x'}$

where $x'$ has no complete group of $n$ identical prime factors left under the root, and $y'$ folds together everything that was pulled out with the original $y$. When $n = 2$ the index is dropped and the radical is written simply $\sqrt{x}$; every other index from $3$ upward is written explicitly, e.g. $\sqrt[3]{x}$, $\sqrt[5]{x}$.

Simplifying, step by step

To simplify $3\sqrt{8}$ by hand:

1. Set the outer coefficient aside for a moment and simplify the bare radical: $\sqrt{8}$.
2. Factor the radicand into primes: $8 = 2^3$.
3. Group into sets of the index (here, 2): one complete pair of 2s, plus a single leftover 2.
4. Pull the complete pair out, taking its square root: $2^2 \to 2$, becoming a coefficient — so $\sqrt{8} = 2\sqrt{2}$.
5. Multiply that extracted coefficient by the original outer coefficient: $3 \times 2 = 6$, giving the final answer $3\sqrt{8} = 6\sqrt{2}$.

This calculator shows the same breakdown for whatever coefficient, radicand, and index you enter — including negative radicands, where the sign either folds into the coefficient (odd index) or attaches an "$i$" (even index) before the outer coefficient multiplies in.

The even/odd sign rule

Whether a negative radicand produces a real or an imaginary answer depends only on whether the index is even or odd — the outer coefficient never changes this, it just multiplies into whichever case applies:

Even index, negative radicand → imaginary. No real number raised to an even power can ever be negative, so $\sqrt[n]{x}$ has no real value when $n$ is even and $x < 0$. The answer carries the imaginary unit $i = \sqrt{-1}$ instead: $\sqrt{-240} = \sqrt{240} \times i = 4\sqrt{15}\,i$. Multiplying by an outer coefficient — say $y = 3$ — simply scales it: $3\sqrt{-240} = 12\sqrt{15}\,i$.

Odd index, negative radicand → real. An odd power preserves the sign of its base, so every real radicand has exactly one real $n$th root when the index is odd, and the sign folds into the coefficient: $\sqrt[5]{-32} = -2$, because $(-2)^5 = -32$ exactly. An outer coefficient of $3$ gives $3\sqrt[5]{-32} = -6$.

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 $2^5 = 32$. When that happens, the outer coefficient multiplies straight into a plain integer, with no radical sign left at all:

Exact form vs. decimal approximation

Most radicands are not perfect $n$th powers, so the exact simplified answer keeps part of the value under the root — an irrational number that never terminates or repeats as a decimal. This calculator shows both forms side by side: the exact form (the simplified radical, e.g. $6\sqrt{2}$) and a rounded decimal approximation beneath it (e.g. $\approx 8.485281$), so you can use whichever one a problem calls for. A complex (imaginary) result has no real decimal value, so only the exact form is shown for those. For a bare, coefficient-free root without the simplification step, see the Square Root calculator (for exact and decimal square roots of any number, including non-integers).

Worked examples

Example 1 — no outer coefficient beyond 1: $\sqrt{44} = 2\sqrt{11} \approx 6.63325$, since $44 = 2^2 \times 11$.

Example 2 — coefficient multiplies the extracted factor: $3\sqrt{8} = 3 \times 2\sqrt{2} = 6\sqrt{2}$, since $8 = 2^3$ pulls out a single pair of 2s.

Example 3 — even index, negative radicand (imaginary): $\sqrt{-240} = 4\sqrt{15}\,i$, since $240 = 2^4 \times 3 \times 5$ and no real number squared can be negative.

Example 4 — odd index, higher root: $\sqrt[3]{54} = 3\sqrt[3]{2}$, since $54 = 2 \times 3^3$ pulls out one complete group of three 3s.

Related roots calculators

This calculator's core engine — pulling perfect $n$th-power factors out of a radicand by prime factorization — is shared with the Radicals calculator (a bare $\sqrt[n]{x}$, no outer coefficient, and it also accepts decimal radicands), the Square Root calculator (fixed index 2), and the Cube Root calculator (fixed index 3). A simplified radical like $2\sqrt{11}$ often shows up as the irrational root term inside a Quadratic Formula result, where $b^2 - 4ac$ under the square root frequently needs exactly this kind of simplification.