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# Radical|Definition & Meaning

## Definition

The general term for the nth root of a value or expression. The symbol for the radical is $\sqrt{}$, usually used as $\sqrt[n]{a}$, where n is the index and a is the radicand. For example, the term $\sqrt[4]{16}$ represents the quadruple root (or 4th root) of the number 16 (the radicand), which equals two since 2$^4$ = 16.

In **mathematics,** the symbol for a **radical** is the under-root, which is also known as the root sign.

A radical is indeed the inverse of an exponent. It may represent a **square** root or cube **root,** and the number that comes **before** the sign or **radical** is **referred** to as the degree or **index** number. This number, a complete number **rendered** as an exponent, **eliminates** the **radical previously** there.

## What Exactly Does the Term Radical Mean?

There is no **difference between** a **number’s radical** and its root; they are the same thing. It’s **possible** that the root is just a square root, a cube root, or even just an nth root. As a result, a radical is a name given to any number or phrase that makes use of a root.

The **English** word **“radical” originates** from the Latin term **“radix,”** which literally translates to “root.” The term “radical” can be used to refer to a variety of distinct roots for a number, such as the square root, the **cube** root, the fourth root, and on.

The number that comes even before radical is called the **number** index or degree, which has a specific meaning. **Multiplying** this number by itself this many times gives us the answer to **how** many times we need to multiply the radical by this number.

This is seen as the **counterpart** of an exponent, much in the same way as addition is the contrary of **subtraction** & division is the opponent of multiplication.

## Representation of Radical

A radical is represented by the symbol as √. A radical is also related to the following terms:

A radical equation involves the variable inside a radical.

The term “radical

**expression”**refers to an**expression**that is contained within a square root.The term “radical inequality” refers to an

**inequality**that is contained within a radical.

The illustration below represents the 4 in terms of the radical sign.

## Basic Radical Principles

The **following** are some broad **guidelines** to follow if you **identify** as a radical:

If the

**number**that is being**radicalized**has a positive value, then the result will also have a positive value.If the

**number**that is being**subtracted**by the radical is negative, then the outcome will also be negative.Irrational

**numbers**are**produced**whenever the number underneath the radical**equals**negative, and also, an**index**is an even integer. This**combination**produces an**irrational**outcome.The

**value**of the radical will equal the**square**root if an**index**also isn’t specified.It is

**possible**to**multiply**numbers with identical radicals and indices.For numbers that fall under the same

**radical,**the division is possible.When the

**number**is divided under the same**radical,**the**multiplication**rule can be**applied**in the other**direction.**In any equation, the

**radical**can also be**expressed**in its exponent form.The radical

**expression**itself is equal to the radical’s inverse exponent, which is the**index number.**

## The Radical Formula

In **order** to get a solution for an equation **containing radicals,** it must first be made radical-free. We have to power both of the equation’s sides with n in order to make an **equation** involving the nth **root** radical free. This **made** the radical equation free of radicals more difficult to see. Let’s **take** a look at the radical **formula** that’s down here.

\[n\sqrt{x} = p\]

\[x^\frac{1}{n} = p \]

\[(x^\frac{1}{n})^n = p^n \]

\[x = p^n\]

- The value ‘n’ is referred to as the index.
- The term “radical” refers to the expression as well as a variable that is located within the radical sign, which in this case is the letter x.

## Application of Radical in Real Life

In **addition** to their usage in a wide number of **mathematical** applications, radicals also have a great many **applications** in the real world. The following are some examples:

When it

**comes**to the solution of equations,**radicals**can be utilized to solve equations that contain roots.Radicals are

**frequently**employed in geometry to**determine**a line segment’s**length**or a**shape’s**area.**Radicals**are**utilized**in a variety of**scientific**and**engineering**domains in order to find solutions to challenges**involving**a range of dimensions and**quantities.**In the field of

**computing, radicals**are used for**operations**on various data structures and algorithms.

In summary, **radicals** have several practical applications in **diverse** domains of mathematics, physics, **engineering,** and **computer** programming. These **applications** can be found in a wide variety of contexts. Equations are solved with them, **dimensions** and quantities are **determined** with them, and statistical **measurements** are **computed** with them.

## Summary

A **mathematical** expression or a **number** can be **represented** by the symbol known as a radical, which stands for the **expression’s** or number’s root. The symbol for it is **termed** as a radical sign and **represented** by √. As an illustration, the **square-root** of 9 is written as the symbol **9**, and even the **27 cube root** is written as the **symbol** $\sqrt{27}$.

**Radicals** can be employed to **symbolize** either positive or negative roots, **depending** on the context. For **instance,** the four **square** root is two, and the **negative four** square root is a **negative** four raised to the **power minus** four, which is equivalent to **two imaginary** units.

**Simplifying** radicals can be **accomplished** by **relocating** any **components** that were **previously** located within the **radical** sign to a new **location** outside of the **radical sign** in the form of a coefficient.

## A Numerical Example of Radical

### Example 1

**Write** the given **number** in terms of radical

\[3^{\frac{6}{2}}\]

### Solution

**Given** that:

\[3^{\frac{6}{2}}\]

We have to **write** it in **radical** form.

**So**:

\[3^{\frac{6}{2}} = (3^6)^{\frac{1}{2}}\]

\[3^{\frac{6}{2}} = 729^{\frac{1}{2}}\]

\[3^{\frac{6}{2}} = \sqrt{729}\]

### Example 2

**Write** the given **number** in **terms** of radical

\[3^{\frac{8}{2}}\]

### Solution

**Given** that:

\[3^{\frac{8}{2}}\]

We **have** to write it in **radical** form.

**So**:

\[3^{\frac{8}{2}} = (3^8)^{\frac{1}{2}}\]

\[3^{\frac{8}{2}} = 6561^{\frac{1}{2}}\]

\[3^{\frac{8}{2}} = \sqrt{6561}\]

### Example 3

**Write** the given **number** in terms of **radical**

\[3^{\frac{7}{2}}\]

### Solution

**Given** that:

\[3^{\frac{7}{2}}\]

We **have** to write it in **radical** form.

**So**:

\[3^{\frac{7}{2}} = (3^7)^{\frac{1}{2}} \]

\[3^{\frac{7}{2}} = 2187^{\frac{1}{2}} \]

\[3^{\frac{7}{2}} = \sqrt{2187} \]

*All mathematical drawings and images were created with GeoGebra.*