Root calculator
Square, cube and nth roots — as a simplified radical first and a decimal second. √72 is 6√2 before it is 8.485281374, and the factorisation that gets you there is shown underneath.
By Alex Seote, Built and maintains Rule Calculator
Checked against the NIST DLMF · How we check
Exact answer
6√2
As a decimal: 8.485281374 — and it never terminates or repeats. 6√2 is the answer that can be written exactly; the decimal is a rounding of it.
How the radical was simplified
Prime factors: 72 = 2³ × 3²
Every complete group of 2 identical factors comes out from under the square root as one copy of that factor. Whatever is left over cannot come out, and stays inside.
Out: 6 · left inside: 2
Why the decimal is the wrong answer to hand back
A root calculator that prints only a decimal has quietly answered a different question. The decimal cannot be simplified further, cannot be added to another radical, and is not equal to the thing it approximates. Where the question is homework, the mark is for the exact form. Where the question is engineering, the exact form is what survives the next three operations without accumulating error.
The other half of the gap is signs. An odd root of a negative number is an ordinary real number, and the standard implementation — raise to the power 1/n — returns NaN for it in every language that follows IEEE 754. That is an implementation detail leaking into an answer, and it is why so many tools claim the cube root of −8 does not exist.
Questions people actually ask
- What does it mean to simplify a radical?
- To pull every factor out from under the sign that can come out. Break the number into primes, and each complete pair under a square root — or each complete triple under a cube root — leaves as a single copy. 72 is 2 × 2 × 2 × 3 × 3, which is one pair of 2s and one pair of 3s, so a 2 and a 3 come out as 6 and a single 2 stays behind: 6√2. Nothing is lost and nothing is rounded, which is the point.
- Why not just use the decimal?
- Because it is not the same number. √2 is irrational: the digits never terminate and never repeat, so any decimal you write is an approximation. Carry 1.414 through three more steps and the error compounds; carry √2 and it stays exact until the last line. Exams ask for simplest radical form for exactly this reason.
- What is the cube root of a negative number?
- A negative number. −2 × −2 × −2 = −8, so the cube root of −8 is −2, and every odd root behaves the same way. Most calculators return an error here, because the usual implementation raises the number to the power 1/3 and IEEE 754 floating point makes a negative base with a fractional exponent NaN. The sign is carried separately on this page, so the answer comes back.
- And an even root of a negative number?
- There is no real answer, and this page says so rather than printing an error code. No real number multiplied by itself gives a negative, because the two signs always cancel. The answer exists in the complex numbers — √−16 is 4i — and the page names it so you know what you are looking at.
- How do I know whether a root is rational?
- It is rational exactly when the simplification leaves nothing under the radical. √144 comes out as 12 with an empty radical, so it is rational; √72 leaves a 2 behind, so it is irrational and no fraction equals it. The page states which of the two you have rather than leaving it to be inferred from a decimal that looks like it might terminate.