Root Calculator

ⁿ√x ⁿ√(mⁿ · r) = m · ⁿ√r ⁿ√x = x^(1/n)

x

The radicand, the number being rooted. It is a whole number here, and it may be negative when the index is odd. The restriction to whole numbers is about the exact form rather than about the arithmetic: the square root of 2.5 is a perfectly ordinary number, but its exact form is a fraction with a radical in it, and that is a different shape of answer from the one this column prints

n

The root index, from 2 to 12. An index of 1 is the identity — the first root of anything is itself — and an index of 0 is undefined, so both are refused rather than answered. The upper bound of 12 is practical: the exact form works by finding the largest perfect nth power dividing the radicand, and those get sparse quickly, so beyond 12 that column would mostly hold an empty radical beside a huge number

ⁿ√x

The root as a decimal, the primary reading, printed to six decimals when it is not exact. It is the value to compute with, and it is the one that loses information: 8.485281 does not tell you that it came from √72 rather than from √71.9999, and the exact form beside it is what does

m · ⁿ√r

The exact form: whatever perfect nth power divides the radicand is taken out of the radical and written as a coefficient in front of a smaller radical. For √72 this gives 6√2, because 72 is 36 × 2 and 36 is a perfect square. When the radicand is a perfect power there is nothing left over, the radical disappears entirely, and the answer is written as a whole number — which is how the page reports that the root came out evenly

x^(1/n)

The same root written as a power, which is how it is computed. The two notations are the same operation and it is worth knowing they are: the exponent 1/n is why a fractional exponent is a root rather than something unfamiliar. The exponent calculator takes that form directly, and it is the page to use when the radicand is not a whole number or the index is not an integer

√

The radical sign with no index written on it always means the square root, which is why the page prints a plain √ for index 2 and ∛ for index 3. Past 12 the notation runs out of single characters, so higher indices are written with the digits in front of the sign, and the page puts a multiplication sign between a coefficient and an index of 4 or more — 2 × 4√2 — because 24√2 would read as twenty-four times the square root of two

Use it when a root has to be simplified rather than only evaluated, which is what makes it different from a decimal-only square root tool: simplifying √72 to 6√2, checking whether a number is a perfect square, working through a problem where the answer is expected in exact form. It is also the page for the two questions that come up around roots in general — what is the cube root of a negative number (a real number, for odd indices), and what is the fourth root of something. Reach for the exponent calculator when the index is not a whole number, since a fractional exponent is the same operation written as a power and that page takes it directly. Reach for the quadratic formula calculator when the radical appears under a formula rather than on its own, since the discriminant is a square root and that page reads its sign as well as its value. Reach for the square root calculator or the cube root calculator when only one of the two indices is wanted, since each of those carries the material specific to its own case.

1. The square root of 16

   1. Index 2, radicand 16 — a square root
   2. 16 is 4 × 4, so it is a perfect square and the whole thing comes out
   3. The root is 4, and it is exact rather than rounded
   4. The exact form is the bare whole number 4, with no radical left at all

   The case the page is most often opened for, and the one that shows how the exact column reports success: no radical, no decimal, just the integer. A reader who only wanted to know whether 16 is a perfect square has the answer in the second column without reading the first.

2. The square root of 72

   1. 72 is not a perfect square, so the root is irrational
   2. The largest perfect square dividing 72 is 36, since 72 = 36 × 2
   3. Taking 36 out of the radical gives 6 outside it: √72 = 6√2
   4. The decimal value is 8.485281 to six decimals, and 6√2 is the exact value

   The example that justifies the whole page. 8.485281 is an approximation and the panel says so by printing the exact form next to it; a competing page that shows only the decimal has left the reader with a number that is not the square root of 72. The leftover 2 under the radical is also the quickest read of the situation — something is still in there, so nothing came out evenly.

3. The cube root of 54

   1. Index 3, radicand 54 — a cube root
   2. 54 is not a perfect cube, but 27 divides it: 54 = 27 × 2
   3. Taking 27 out gives 3 outside the radical: ∛54 = 3∛2
   4. The decimal value is 3.779763

   The same simplification with a different index, and the case that shows the page is one tool rather than two. The step that has to be done by hand is finding the largest perfect cube below 54 — 27, not 8 — and the reference table on this page is exactly the list that makes that lookup quick: three cubed is 27, four cubed is 64, and 64 is too big.

4. The fourth root of 32

   1. 32 is 16 × 2, and 16 is a fourth power, since 16 = 2⁴
   2. Taking 16 out of the radical leaves 2 outside: ⁴√32 = 2 × ⁴√2
   3. The multiplication sign is written because an index of 4 has no single-character radical sign
   4. The decimal value is 2.378414

   Where the notation has to be careful. Writing this as 24√2 would be read by most people as twenty-four times the square root of two, so the page inserts a multiplication sign whenever the index is 4 or above — the coefficient and the index are both numbers and something has to separate them. Neither √ nor ∛ has this problem, which is why the sign appears only from index 4 upward.

5. The cube root of −54

   1. The radicand is negative and the index is odd, so the root is a real number
   2. The magnitude simplifies exactly as before: ∛54 = 3∛2
   3. The sign is carried outside, and the exact form prints it at the very front: −3∛2
   4. The decimal value is −3.779763

   The negative case, and the one place where the exact form has a rule of its own: the minus sign goes at the front of the whole expression rather than inside the radical. It is also the entry point the exponent calculator sends people to — that page refuses a negative base with a fractional exponent because the arithmetic behind it goes through a logarithm, and this is the page that handles the same question by taking a root instead.

The radicand must be a whole number. The square root of 2.5 is a real number and this page will not take it, because the exact form of that root is a fraction with a radical over it — a shape the exact column has no room for — and printing only the decimal there would hide the fact that the exact answer had been dropped. Any fraction or decimal radicand belongs on the exponent calculator instead, entered as a fractional power. Negative radicands are accepted for odd indices only: the cube root of −8 is −2, but the square root of −4 is not a real number and this page refuses it rather than printing a complex value. The root index is a whole number between 2 and 12; an index of 1 is the identity and an index of 0 is undefined, so neither is accepted, and fractional indices go to the exponent calculator as fractional powers. Roots that do not come out evenly are printed to six decimals, which is a display limit rather than a precision one — the arithmetic behind them is full double precision. The exact form is simplified as far as pulling out perfect powers, which is the standard form of a simplified radical, and nothing here rationalizes a denominator: a root sitting under a fraction is the reader's to deal with. The radicand is bounded at a billion, and that bound is about cost rather than mathematics — the simplification works by trial division, and a bigger radicand would make the page slow to respond to typing rather than wrong.

What is the difference between the exact form and the decimal?

The decimal is a rounded value and the exact form is the value itself. The square root of 72 is 6√2 exactly; 8.485281 is what that comes to in six decimals, so multiplying 8.485281 by itself gives 71.99999… rather than 72. The page prints both because either one alone is misleading: the decimal is the one you compute with, and the exact form is the one that tells you whether the answer came out evenly.

How do I simplify a square root?

Look for the largest perfect square that divides the number, take its root out front, and leave the rest inside. For 72 that is 36, because 72 = 36 × 2, so √72 = 6√2. If no perfect square divides the number — which is the case for 2, 3, 5, 7 and every other number with no repeated prime factor — the radical cannot be simplified and the exact form is just √2.

Can I take the square root of a negative number?

Not on this page, and not as a real number anywhere. The square root of −4 is 2i, a complex number, and this page is a real-number tool. Odd indices are different: the cube root of −8 is −2, a perfectly ordinary real number, because multiplying three negatives gives a negative. So the page accepts −8 with an index of 3 and refuses −4 with an index of 2.

Why can't I enter a decimal as the radicand?

Because the exact form has nowhere to put it. The square root of 2.5 is √10 ÷ 2 — a fraction with a radical in the numerator — and this page's exact column prints a coefficient in front of a radical, not a fraction. Rather than print the decimal and drop the exact answer silently, the page restricts the radicand to whole numbers and says so. A decimal radicand can be entered as a fractional power on the exponent calculator.

What does the index on the radical sign mean?

It is how many times the root has to be multiplied by itself to give the radicand back: an index of 3 means a cube root, so ∛8 is 2 because 2 × 2 × 2 = 8. No index written means 2, which is why this page prints a plain √ for square roots and ∛ for cube roots. An index of 1 would return the number unchanged and an index of 0 has no meaning, so the page starts at 2.

Is a cube root the same as raising to the power of one third?

Yes, and that is how the page computes it. The nth root of x is x to the power of 1/n, so the cube root of 54 is 54 to the power of 0.3333…, which is 3.779763. The two notations describe one operation. They part company on negative bases: the exponent form goes through a logarithm and would refuse −54, while taking an odd root of it is straightforward, which is why the exponent calculator sends those cases here.

• nth Root — the definition of an nth root, including the rule that an even root of a negative number has no real value and an odd one does — Wolfram MathWorld (United States)
• Square Root — the index-2 case, with the simplification of radicals and the table of perfect squares this page's reference table extends — Wolfram MathWorld (United States)
• Cube Root — the index-3 case, and the reason a negative radicand has a real cube root where it has no real square root — Wolfram MathWorld (United States)