Radicals in Maths
In this article we explain what radicals are in the mathematical context, some examples, properties and the operations that can be performed between them.
Table of Contents
What are radicals?
In mathematics, radicals are a notation used to represent the roots of a number or an algebraic expression.
The most common symbol for representing a radical is the principal root symbol *\sqrt{~}*, which is used for the square root. For example, *\sqrt{9}* represents the principal square root of 9, which is equal to 3, since *3^2=9.*
The names of each part of the radicals are:
- Index: It is the number indicating the type of root being taken. For example, in the square root, the index is 2, in the cube root, the index is 3, and so on.
- Radical symbol: It is the symbol *(\sqrt{~})* representing the operation of finding the root.
- Radicand: It is the number under the radical sign, from which the root is sought.
Radicals can be simple, like the square root or cube root of a specific number, but they can also be more complex, involving other algebraic expressions. For example, *\sqrt{x^2 + 2x + 1}* represents the square root of the expression *x^2+2x+1.*
Radicals can be used to represent a variety of quantities, such as lengths, areas, and volumes. For example, the square root of an area is the side of a square with that area. The cube root of a volume is the side of a cube with that volume.
Examples of radicals
Here are some examples of radical expressions: *\sqrt{7};* *~~\sqrt[3]{-13};* *~~\sqrt{x^2 + 2x + 1};* *~~\sqrt[3]{5};* *~~\sqrt[4]{\dfrac{x^3 +3x^2}{x-1}};* *~~\sqrt[7]{6};* *~~\sqrt{9x};* *~~\sqrt[4]{-3x+1};* *~~\sqrt[3]{\dfrac{-6z}{x}};* *~~\sqrt{16x^2-9y^2}.*
It is important to note that roots with an even index of negative numbers do not exist in the field of real numbers; for example: *\sqrt{-1}* is not a real number. Instead, roots with an odd index of negative numbers do exist, for example, *\sqrt[3]{-8}.*
Operations with radicals
Basic operations with radicals can be performed as long as certain conditions are met. It is useful to know the properties of radicals as they allow for simpler problem-solving.
Simplification and rationalization
Rationalization is a process that involves removing radical expressions either from a numerator or a denominator. Simplification involves finding the simplest form of writing the same radical.
FAQs
What is a radical in mathematics?
A radical is an expression that represents a root of a number or an algebraic expression. The radical consists of three parts: the radical symbol (√), the index (indicating the type of root) and the radicand (the number or expression under the radical symbol).
What is the difference between a positive and a negative radicand?
The radicand is the number or expression under the radical symbol. The difference lies in the possibility of obtaining a real root. When the radicand is positive, a real root can be obtained, as in the case of the square root of a positive number. However, if the radicand is negative and the root index is even, there is no real root in the set of real numbers.
Which radicals do not exist?
In the set of real numbers, even roots of negative numbers do not exist. For example, the square root of a negative number does not exist, because the square of any real number is always non-negative.
