Square Root Calculator

√x = √(m² · r) = m · √r √(n ÷ d) = √(n · d) ÷ d

x

The radicand, the number under the radical sign. It cannot be negative: a negative radicand has no real square root, and squaring any real number gives a non-negative result, so the page refuses it and points at the odd-index roots on the general root calculator instead. Zero is fine — the square root of 0 is 0.

m²

The largest perfect square that divides the radicand: 4, 9, 16, 25, 36 and so on. Finding it is the whole of the simplification, and the page searches from the top down so that the first factor it finds is the largest one. For 72 that factor is 36, which leaves 2 behind under the radical.

m

The square root of that perfect square, which comes out in front of the radical sign: 6 in the case of 72. When the coefficient is another whole number it is printed plainly, so the exact form reads 6√2. When it is 1 the coefficient is dropped rather than printed, because 1√2 is not how the number is written.

r

Whatever is left under the radical once every square factor has been pulled out. It is square-free, meaning no perfect square divides it, which is exactly what makes the form final. If nothing is left — if the radicand was a perfect square to begin with — the radical sign disappears entirely and the exact form is just a number, as with the square root of 144.

√x

The decimal value, reported to six places. It is the same number as the exact form evaluated — 6√2 is 8.485281 — and it is printed beside it rather than instead of it, since the two answer different questions. The decimal is what a reader measures with; the exact form is what a reader proves with.

1. Simplifying the square root of 72

   1. Look for the largest perfect square that divides 72
   2. 36 divides it: 72 ÷ 36 = 2, so 72 = 36 × 2
   3. The square root of 36 is 6, which comes out in front: √72 = 6√2
   4. Evaluating the exact form gives 8.485281

   The case that makes the two-column layout worth having. As a decimal, √72 is 8.485281 and there is nothing more to say about it; as an exact form it is 6√2, which shows at a glance that squaring it gives 72 back exactly. The check is one line: 6√2 squared is 36 × 2, which is 72.

2. A radicand with one decimal place

   1. A decimal radicand is turned back into a fraction first: 1.5 = 3/2
   2. Split the root across numerator and denominator with √(n ÷ d) = √(n · d) ÷ d
   3. That gives √(3 × 2) ÷ 2 = √6 ÷ 2
   4. The decimal is 1.224745

   The exact form uses a division sign rather than a slash on purpose. Written as √6/2 the line is ambiguous — it reads just as easily as the square root of six halves, which is √3. The page prints √6 ÷ 2 so there is only one reading. This is the case the general root calculator deliberately does not take, since it requires whole radicands.

3. A decimal that turns out to be a perfect square

   1. 2.25 as a fraction is 9/4
   2. Both parts are perfect squares: √9 = 3 and √4 = 2
   3. Nothing is left under the radical, so the exact form is the fraction 3/2
   4. As a decimal it is 1.5

   When the radical empties out, the exact form stops being a radical at all — it is a plain fraction, and that is the signal that the radicand was a perfect square. The exact form 3/2 is worth preferring to 1.5 in any later arithmetic, since multiplying by 3 and dividing by 2 keeps everything exact where the decimal would not.

The radicand may not be negative: the square root of a negative number is not a real number, and squaring a real number never produces one. Zero is accepted and gives zero. The radicand is limited to two decimal places, and a third decimal is refused rather than rounded — the simplification works by turning the decimal into a fraction and then trial-dividing, and every extra decimal place multiplies the work by ten, so the page states the limit instead of quietly printing a simplified form that belongs to a slightly different number. The exact form is a notation, not a number: it is written the same way in every language, since a radical sign and a division sign carry no words. The decimal, by contrast, is rounded to six places, which is a display limit rather than a precision one. The page has a fixed index of two — it simplifies square roots only. Cube roots, fourth roots and any other index, including the odd-index roots of negative numbers, belong on the general root calculator this page links to.

What does simplifying a square root actually do?

It moves the largest perfect square out from under the radical sign. The square root of 72 is the same number as 6√2 — nothing has been approximated or rounded — but 6√2 is the tidier way to write it, and it is the form that stays exact when it is used in later arithmetic. Simplifying never changes the value, only the spelling.

How do I know when a square root is fully simplified?

When nothing is left under the radical that a perfect square divides. In 6√2 the 2 is square-free: 4 does not divide it, 9 does not, and no larger square does either, so the form is final. That is also how the page decides when to stop — it searches for the largest square factor, from the top down, and stops at the first one it finds.

Why is the exact form written with a division sign?

Because the slash would be ambiguous. √6/2 reads as either the square root of six, divided by two, or the square root of six halves, and those are different numbers — the second one is √3. Printing √6 ÷ 2 leaves one reading. It is the same reason the coefficient is dropped when it is 1: the notation is chosen so that it cannot be misread, not so that it is short.

Can I enter a decimal radicand?

Yes, up to two decimal places. The page turns the decimal back into a fraction, then takes the root of the numerator and the denominator separately using √(n ÷ d) = √(n · d) ÷ d. The square root of 1.5 is √6 ÷ 2, and the square root of 2.25 is exactly 3/2. A third decimal place is refused, because each extra place multiplies the trial division by ten.

Why does the square root of a decimal sometimes come out as a fraction?

When the radicand is itself a perfect square of a fraction, nothing is left under the radical and the exact form collapses to a plain fraction — the square root of 2.25 is 3/2. This is the same rule as a whole-number radicand that is a perfect square, where the exact form collapses to a plain integer; if the radical empties, there is nothing left to write it around.

Why is a negative radicand refused here?

There is no real number that squares to a negative number, so the square root of a negative radicand is not a real number at all. Every square root on this page has a fixed index of two, so there is no odd-index half where negatives would be allowed. The general root calculator does take cube roots of negative numbers, because an odd power of a negative number is still negative, and this page points there.

• Square Root — the definition, the principal root, and the rule for pulling square factors out of a radicand — Wolfram MathWorld (United States)
• Radical — the radical sign, the radicand, and the convention that the principal square root is the non-negative one — Wolfram MathWorld (United States)
• Square Number — the perfect squares the simplification searches for, from 4 up to whatever the radicand allows — Wolfram MathWorld (United States)