A day full of math games & activities. Find one near you.

Radical

In maths, a radical is the opposite of an exponent that is represented with a symbol '√' also known as root. It can either be a square root or a cube root and the number before the symbol or radical is considered to be an index number or degree. This number is a whole number represented as an exponent that cancels out the radical. Let us learn more about the concept of radicals, the formula, and solve a few examples.

1.	What is Radical?
2.	Radical General Rules
3.	Radical Formula
4.	FAQs on Radical

What is Radical?

The radical of a number is the same as the root of a number. The root can be a square root, cube root, or in general, n^th root. Thus, any number or expression that uses a root is known as a radical. The term radical is derived from the Latin word Radix which means root. The radical can describe different kinds of roots for a number such as square root, cube root, fourth root, and so on. The number written before the radical is known as the index number or degree. This number helps in telling us how many times the number would be multiplied by itself to equal the radicand. This is considered to be the opposite of an exponent just like addition being the opposite of subtraction and division being the opposite of multiplication. For example: ∛125 = 5 as 5 × 5 × 5 = 125.

Radical Definition

The symbol '√' that expresses a root of a number is known as radical and is read as x radical n or n^th root of x. The horizontal line covering the number is called the vinculum and the number under it is called the radicand. The number n written before the radical is called the index or degree. Some examples of radicals are √7, √2y+1, etc.

A radical can also be associated with the following terms:

An equation that is inside a radical is known as a radical equation.
An expression that lies inside a square root is known as a radical expression.
An inequation that is inside a radical is known as radical inequality.

Radical General Rules

Mentioned below are a few general rules for a radical.

If the number is positive under the radical, the result will be positive.
If the number is negative under the radical, the result will be negative.
If the number under the radical is negative and an index is an even number, the result will be an irrational number.
If an index is not mentioned, the radical will be square root.
Multiplication of numbers under the same radical and index is possible. For example, ∛12 × ∛10 = ∛120.
Division is possible for numbers under the same radical. For example, √8/√4 = √8/4 = √2.
The reverse of the multiplication rule is possible, where the number is split under the same radical. For example, √27 = √9 × √3 = ∛3 × √3.
The radical can be written in its exponent form as well in any equation. For example, √x = 25 (√x)² = (25)² x = 5.
The inverse exponent of the index number is equivalent to the radical itself. For example, √7 = (7)^1/2.

Radical Formula

To solve a radical equation, it has to be made radical-free. To make an equation of n^th root radical free, we power both sides of the equation with 'n'. This masked the radical equation-free from radical. Let's look into the radical formula below.

n√ x=p

x^1/n = p

(x^1/n)ⁿ = pⁿ

x = pⁿ

where,

The n√ symbol is known as the radical of n^th root.
'n' is known as the index.
The expression or variable inside the radical symbol i.e, x is known as the radicand.

Radical Formula

Let's take some examples to understand the radical formula.

☛Related Topics

Listed below are a few topics that are related to a radical.

Radical Examples

Example 1: Express 3^3/2 in radical form using radical formula.

An expression that uses a root, such as a square root, cube root is known as a radical notation.

As per the Power Rule of Exponents,

(a^m)ⁿ= a^mn

Thus, 3^3/2 can be written as (3^1/2)³

=> (3^1/2)³ = √3³ (since, √x is expressed as x^1/2)

Now to express in radical form using the radical formula, we must take the square of the number in front of the radical and placing it under the root sign:

3^3/2= 3 √3 = √3 ³ = √27

Therefore, 3^3/2 in radical form is √3³ = √27.
Example 2: Solve the radical: \(\sqrt[3]{x}\) = 8 using the radical formula.

Solution:

Given:

\(\sqrt[3]{x}\)= 8

To make it radical free, using radical formula

x^1/3 = 8

(x^1/3)³ = 8³

x = 8³

x = 512

Therefore, Thus, the value of x for \(\sqrt[3]{x}\) = 8 is 512.
Example 3: Solve the radical expression (6 + 3√x)/y. Where x = 16 and y = 2.

Solution:

Given,

x = 16 and y = 2.

Solving the radical expression,

(6 + 3√x)/y

(6 + 3√16)/2

(6 + 3×4)/2

= 9

Therefore, the radical expression is 9.

go to slide go to slide go to slide

Great learning in high school using simple cues

Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes.

Book a Free Trial Class

FAQs on Radical

What is the Meaning of Radical in Math?

Radical is the symbol '√' that helps in identifying the n^th root of a number i.e. square root, cube root, fourth root, etc. The radical or symbol means the root of any number. The horizontal bar across the number i.e. is the radical should cover the entire number to conduct any of the arithmetic operations such as addition, subtraction, multiplication, or division. The number under the radical is called radicand and the number before the radical is called the index.

How Do You Solve a Radical?

A radical can be solved by using any of the rules mentioned above in the article. Numbers under the same radical and index can be multiplied, subtracted, divided, and added.

What is the Radical Formula?

To solve any equation to be radical-free, we use the formula.

x = pⁿ

where,

The n√ symbol is known as the radical of n^th root.
'n' is known as the index.
The expression or variable inside the radical symbol i.e, x is known as the radicand.

What is Exponent and Radical?

An exponent of any number shows how many times we are multiplying a number by itself. Whereas a radical is a symbol used to express the root of any number. A radical can be expressed in an exponential form.

Math worksheets and
visual curriculum