Here you can use our Root Calculator to calculate the n-th root of any number. The n-th root of any number is also called the root degree, and CalCon allows you to calculate it using an interactive calculator on this page or by downloading apps from the Play and App store. It is pretty simple because all you need to enter is the power of root and the number whose root you are looking for.

Below we will give a brief theoretical overview of what a root is in math, how it is estimated and calculated, then how to estimate the square root, a quick overview of what a cubic root is, and more. Also, check out some of our related calculators on this topic to help you with Cube Root or Square Root. Check other calculators as well, like Percentage, Percentage Difference or some calculator related to geometry.

What is a Root in math?

The n-th root of a given real number x (arithmetic) is a number that multiplied by itself n times gives x. The arithmetic operation by which we find the root is called rooting.

If we write that y is the n-th root of x like the equation bellow

y = \sqrt[n]{x}

Then the following equations are valid:

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

And at the end, in terms of logarithms

\log_y(x) = n

In the expression \sqrt[n]{a} , the natural number n is called the exponent, the degree of the root or the index, while the a is called the radicand. Square and cubic roots are most often used, and we will say something about them later.

On a set of real numbers, roots with an even exponent (second, fourth, sixth, etc.) are real only for zero and positive. For negatives, determining the even root requires introducing an imaginary unit (complex numbers).

In algebra, the definition of roots is extended to non-integer exponents and complex numbers. However, the root of a complex number cannot be uniquely defined as its logarithm.

How to find a Root?

Finding roots in math is the inverse problem of raising to a power. There are different roots: second-degree roots, third-degree roots, fourth-degree roots, etc. It depends on the degree to which the number has initially been powered. The root is marked with the symbol: √ is a square root, i.e., a root of the second degree; if the root has a degree greater than the second, the corresponding degree is assigned above the sign of the root. The number under the sign of the root is a radical expression.

Basic rules

When finding the root, there are a few rules to help you make no mistake:

The even root (if degree 2, 4, 6, 8, etc.) of the negative number does NOT exist. If the radical expression is negative, but an odd root is required (3, 5, 7, and so on), the result will be negative.

The root of any power of one is always one: \sqrt{1} = 1 .

The root of zero is zero: \sqrt{0} = 0 .

Estimating a Root

One of the simplest ways to find the square root is to use a particular table. You can use the table to find the square of any number from 10 to 99. In this case, the rows of the table contain the values ​​of tens, in the columns – the values ​​of units. The cell that is at the intersection of the column and row contains a two-digit square. For example, to square 63, you need to find row 6 and column 3. At the intersection, we find cell 3969.

Since extracting the root is the opposite of squaring, you need to do the opposite to perform this action: first, find the cell with the number whose radical you want to calculate, and then determine the answer by the column and row values. As an example, consider calculating the square root of 169.

Similarly, you can find the cubic and n-th degree roots using the appropriate tables.

The convenience of this method is its simplicity and the absence of additional calculations. However, the disadvantages are obvious: the method can only be used for a limited range of values (the number must be in the range of 100 to 9801). In addition, it will not work if the specified number is not in the table.

Estimate a square Root

The most common roots are the square, where n = 2, and the cubic, where n = 3. Calculating square roots and nth roots are fairly intensive. It requires estimation, trial, and error. There are more precise ways to estimate square roots, but we will simplify and present a method that does not require a significant understanding of complicated math concepts. To estimate √a:

  1. Estimate b
  2. Divide a by b. Stop if the c returned is precise to the desired decimal place.
  3. Average b and c and the result is a new guess
  4. Repeat step two

For instance, find √27 to 3 decimal places

First, we need to guess: 5,125, then:

27 \div 5,125 = 5,268

\frac {5,125+5,268}{2} = 5,197

27 \div 5,197 = 5,195

\frac {5,195+5,197}{2} = 5,196

27 \div 5,196 = \mathbf{5,196}

So the result, rounded to three decimal places is 5,196.

Estimate an n-th Root

Calculating nth roots can be done using a similar method, with some modifications regarding n. Estimating higher nth roots is hard, even if using a calculator for intermediary steps. A simple method is given below with an example. To estimate n√a follow the next form:

  1. Estimate b
  2. Divide a by bn-1. Stop if the c returned is precise to the desired decimal place.
  3. Average of equation: [b × (n-1) + c] / n
  4. Repeat step two

Find 8√15 to 3 decimal places

It’s time for guess: 1.432

15 \div 1,432^7 = 1,405

\frac {1,432 \cdot 7+1,405}{8} = 1,388

15 \div 1,388^7 = 1,403

\frac {1,403 \cdot 7+1,388}{8} = 1,402

It should then be clear that continuing any further will result in a number that would round to 1.403, making 1.403 the final estimate to 3 decimal places. To conclude, this is an iterative process.

Square, Cube, fourth, n-th Root

As for the square root, there are several other methods by which the roots can be calculated, and some of the known ones are:

  1. Guess and check method (explained above)
  2. Average method
  3. Repeated subtraction method
  4. Prime factorization method (Read more at the Cube Root Calculator site)
  5. Long Division method
  6. Number line method

The repeated subtraction and prime factorization methods are applicable only for perfect square. Perfect squares are the numbers whose square roots are integers, i.e., 1, 4, 9, etc.

On the other hand, if you need the fourth or n-th roots, best way is by entering values in the calculator.

Using the Root calculator

Let’s do a couple of examples of calculations, so; to begin with, we need to find the second root of the number 98 using our calculator.

Enter 98 in field a, while in field n, you should enter the value 2. The result will be 9.9.

All other variants are calculated similarly, such as the 54th root of the 1745, which is 1.1482.


How to find a square root?

There are several methods, like prime factorization or the repeated substraction, but also you can do it with the calculator.

Where do we need roots and squares?

The concept of square roots and squares are used in various fields of everyday life, such as carpentry, engineering, architecture, technology, etc.

Root of a fraction?

The root of a fraction is equal to the root of the numerator divided by the root of the denominator.