As per the __Power Rule__ of Exponents,

# Radical Formula

The radical of a number is the same as the root of a number, and we will learn more about the radical formula as we go along. 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.

- 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.

Some examples of radicals are √7, √2y+1, etc. Let's look into the concept of radical formula followed by solved examples.

## What is 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} = p^{n}

x = p^{n}

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.

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

## Solved Examples Using Radical Formula

**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.

(a^{m})^{n}= a^{mn}

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

=> (3^{1/2})^{3} = √3^{3} (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

**Answer: **Therefore, 3^{3/2} in radical form is √3^{3} = √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} = 8^{3}

x = 8^{3}

x = 512

**Answer: **Thus, the value of x for \(\sqrt[3]{x}\) = 8 is 512.