Nth Root Calculator
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Other tools you may find usefulSquare root as a basic mathematical operation
Square root is the inverse operation of power, used to find a number that, when raised to a specific power, gives a subroot number. The most popular type of root is the square root (of the second degree, marked with the symbol √), which answers the question: what number multiplied by itself gives a given value. The second most frequently used is the cube root (of the third degree, ³√), geometrically related to the length of the edges of a cube with a given volume. Square roots are essential in algebra, trigonometry, geometry (e.g. in calculating diagonals, areas and volumes), and in physics and engineering.
Our free online radical calculator allows you to extract 2nd, 3rd and any radical roots from positive numbers. In the case of numbers from which it is impossible to take a perfect square root, the calculator provides exact decimal rounding and a simplified symbolic form.
Definition and mathematical formula of the root
Formally, the root of the degreenof the numberais defined as follows:
n√a = b <=> bn= a
where:
- nis the degree of the root (for n = 2 degree is usually omitted in the notation),
- ais the square root number (for an even degree it must be a non-negative number),
- bis the result of the square root.
Roots can also be written as a power with a fractional exponent:
n√a = a1/n
Table of square and cube roots for common numbers
The table below lists the values of second (√a) and third (³√a) roots for numbers, which give the total results:
| Square root (a) | Square root (√a) | Reason (b² = a) | Cube root (³√a) | Reason (b³ = a) |
|---|---|---|---|---|
| 1 | 1 | 1² = 1 | 1 | 1³ = 1 |
| 4 | 2 | 2² = 4 | ≈ 1.59 | 1.59³ ≈ 4 |
| 8 | ≈ 2.83 | 2.83² ≈ 8 | 2 | 2³ = 8 |
| 9 | 3 | 3² = 9 | ≈ 2.08 | 2.08³ ≈ 9 |
| 16 | 4 | 4² = 16 | ≈ 2.52 | 2.52³ ≈ 16 |
| 27 | ≈ 5.20 | 5.20² ≈ 27 | 3 | 3³ = 27 |
| 64 | 8 | 8² = 64 | 4 | 4³ = 64 |
| 100 | 10 | 10² = 100 | ≈ 4.64 | 4.64³ ≈ 100 |
| 125 | ≈ 11.18 | 11.18² ≈ 125 | 5 | 5³ = 125 |
Properties of elements and operations on them
When transforming expressions with radicals, it is worth using the following mathematical identities:
- Square root of the product:
√(a × b) = √a × √b - Square root of the quotient:
√(a / b) = √a / √b - Disabling the factor before the sign of the square root:
√50 = √(25 × 2) = 5√2
Frequently asked questions (FAQ)
What is square root?
This is the inverse operation of exponentiation. It involves finding a number that, when raised to a given power, will give a square root number.
Is it possible to take the square root of a negative number?
In the set of real numbers, roots of even degrees (e.g. square) of negative numbers do not exist. They are defined only in the set of complex numbers (e.g. √-4 = 2i, where i² = -1). Square roots of odd degrees (e.g. third degree) of negative numbers are real numbers (e.g. ³√-8 = -2).
How to calculate square root in Excel?
In Excel, the ROOT function is used for this purpose (e.g.=SQUOT(16)will return 4).
What does it mean to remove the irrational from the denominator?
It is the process of transforming a fraction so that its denominator does not contain a square root (an irrational number). This is done by multiplying the numerator and denominator by the same expression.
How to quickly estimate the value of an element without a calculator?
Find the two closest numbers whose roots are integers. For example, to estimate √30: we know that √25 = 5 and √36 = 6. So √30 lies between 5 and 6 (approximately 5.48).