Radical Calculator

A root is the value that, when multiplied by itself a whole number of times, equals the original number. Select the desired root (square root, cube root, fourth root, etc.) and enter any number. This calculator will return the value of the root, indicate whether or not the number is a power, and convert the result to a simplified radical expression.

The square root is just the starting point of an infinite family of roots. The cube root finds what number multiplied by itself three times gives you the original number, while the fourth root finds what number multiplied by itself four times gives you the original number. And so on. Changing the exponent changes everything. It also changes whether there are real solutions for negative numbers.

What is a root expression?

A radical expression is made up of three parts: The radical symbol is the curved figure at the front. The number in the "mouth" of the radical symbol is the index and tells you what kind of root to take. The value under the bar is called the radicand, which is the number that you are taking a root of.

Symbolically: The nth root of x is a value r which, when multiplied by itself n times, equals x.

$\sqrt[n]{x} = r \iff r^n = x$

So the cube root of 8 is equal to 2 because 2 raised to the third power is 8. The fourth root of 16 is 2 because 2 raised to the fourth power is 16. When a radical expression has no index, it is understood to be 2; that is, the square root. Since every root has an equivalent exponential representation, the following two forms are exactly the same thing.

$\sqrt[n]{x} = x^{1/n}$

Understanding roots is important.

It is helpful to know some perfect powers, that is numbers whose roots are integers. Each series contains numbers worth memorizing.

Looking down the middle column you can see the pattern that this calculator is looking for. If a number is in the list of selected roots then the result will be a clean integer and the calculator will show it as a perfect power.

Simplify roots

Most numbers are not perfect powers, but many can be simplified in terms of their roots. Simplifying a radical expression can often be achieved by factoring out perfect powers. This is because the root of a product can be broken into the product of separate roots.

$\sqrt[n]{a^n \cdot b} = a \, \sqrt[n]{b}$

The trick is to find a factor of the radicand that is itself a perfect power for the chosen root, extract the root of this factor and put it as an integer in front of the remaining number. For example, the cube root of 54 has the hidden perfect power 27 so the root of 54 can be written as: 3 times the cube root of 2.

This simplification process can be represented as follows: $\sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2}$.

Let's look at two more examples. If you extract the factor 64 from the cube root of 192, what remains is 4 multiplied by the cube root of 3. If you extract the factor 36 from the square root of 72, what remains is 6 multiplied by the square root of 2. The factors table at the top of this article breaks down your numbers and extracts complete factors one after another.

Simplification is only possible if the number under the root contains a factor that is a complete power for the corresponding root exponent. The cubic root of 10 cannot be simplified further since 10 is the product of 2 and 5, neither of which are complete cubes. In this case, the calculation tool reports that the root already exists in its simplest form.

Odd roots, even roots, negatives.

This is where a general root differs from a normal square root. Whether or not a negative number has a real root depends entirely on whether the index of the root is odd or even.

Negative numbers with odd roots are negative real numbers. The cube root of -8 is -2 because -2 cubed equals -8. The fifth root of -32 is -2. In contrast, negative numbers with even roots are not real numbers since any real number raised to an even power can never be negative.

This calculator handles signs. If you select an odd root and enter a negative number, it will return a negative real value. If you select an even root and enter a negative number, it will indicate that there is no real solution.

How many roots does a number have?

For positive numbers, even root indices give two real roots, one positive and one negative, which are the same in magnitude. The fourth roots of 16 are +2 and -2, since both raised to the fourth power gives 16. The positive root is called the principal root and is the value shown when only the radical symbol is used.

Odd root indices give exactly one real root which has the same sign as the number. The cube root of 8 is only 2. Similarly, the cube root of −8 is only −2. Zero is a special case since all roots of zero are equal to zero.

Addition, multiplication and division of roots.

Adding roots is different from adding regular numbers. You can only combine terms if they have the same root exponent and radicand, then add the coefficients. So the sum of the product of 2 and the cube root of 3 and the product of 5 and the cube root of 3 is equal to the product of 7 and the cube root of 3. In contrast, the sum of the cube root of 3 and the cube root of 2 cannot be combined at all.

When the roots have the same index, multiplication and division become even easier. You can simply combine the numbers under the radical sign.

$\sqrt[n]{a}\,\sqrt[n]{b} = \sqrt[n]{a b} \qquad \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\tfrac{a}{b}}$

The product is usually something that can be simplified further. The cube root of 4 times the cube root of 16 becomes the cube root of 64 which simplifies to a more manageable 4.

Roots and fractional exponents.

Any root can be converted to an exponent with a fractional power. This is the format used in the calculator tool. An nth root corresponds to a power of 1/n and mixed forms deal with powers within the radical.

$x^{m/n} = \sqrt[n]{x^m} = \left(\sqrt[n]{x}\right)^m$

Thus the two-thirds power of 8 is the cube root of the square of 8, which equals the square of 2, or 4. By considering roots as fractional powers, we can treat both square roots, cube roots and all intermediate forms with the same exponent rules.

Are roots rational or irrational numbers?

A root is rational if and only if the radicand is a perfect power of the index. This is because the value of the root in this case will be an integer. The cube root of 27 is the rational number 3. All other roots are irrational numbers, as they cannot be expressed as a ratio of two integers and their decimal representation goes on forever without repeating.

The square root of 2 is a typical example. The cube root of 2 and the fifth root of 3 are irrational for the same reason. A quick way to determine this is by using the calculator result. If a number is not a perfect power for the chosen root exponent, then the result will be an irrational number.

To these terms

The word radical comes from the Latin radix (root), and so do radishes and eradicate. It has nothing to do with extreme. Here it refers to a root or basis. The horizontal line above the number is called a vinculum, which means bond in Latin, and groups together whatever the root applies to.

The little raised number in the gap of the radical, or index, is the key to expanding a normal square root into a family of complete root terms that this calculator can process.