Here, we will learn completely about the **nth root of a** number like what is it mean, what is the symbol to express the given context, how to find the nth root, and some solved examples of radical expressions. Students who are pursuing 9th Grade Math should understand this concept and answer all types of questions asked in the examinations from the Exponents and Indices. Practice with the worked-out examples and learn the concept thoroughly for grasping the advanced math topics.

## nth root of a – Definition

The principal nth root of a is written as \(\sqrt[n]{ a }\), if a is real number with one nth root. \(\sqrt[n]{ a }\) is the variable or number with the same sign as a that when raises to the nth power, equals a. Here, the index of the radical is n.

\(\sqrt[n]{ a }\) expression means nth root of a. So, (\(\sqrt[n]{ a }\))^n = a

Also, (a^{1/n})^{n} = a^{n x 1/n} = a^{1} = a

Hence, \(\sqrt[n]{ a }\) = a^{1/n}

### Nth Root Symbol

The symbol applied to define the nth root is \(\sqrt[n]{ a }\). It is a radical symbol utilized for square root with a little n to determine the nth root. When it comes to expression ie., \(\sqrt[n]{ a }\), a is known as a radicand, and n is called as an index.

### How to find the nth root of a number?

The calculation of the nth root of a number can be possible by the newton method. Let’s take a look at the below points and understand how to obtain the nth root of a number, A using the Newton method.

Firstly, begin with the initial guess x_{0}, and then repeat using the recurrence relation.

x_{k+1} = \(\frac { 1 }{ n } \)((n-1)x_{k} + \(\frac { A }{ x_{kn+1} } \) ), till the required precision is reached.

Based on the application of nth root, it can be easy to use only the first Newton approximant:

\(\sqrt[n]{ x^{n} + y }\) ≈ x + \(\frac { y }{ nx^{n-1} } \)

### Rational Exponents

Another method to prove principal nth roots is called Rational exponents. The general form for converting a radical expression to radical symbol with a rational exponent is

*a ^{m/n} = (\(\sqrt[n]{ a }\))^{m} = \(\sqrt[n]{ a^{m} }\))*

### Solved Examples on Simplifying Nth Root of A Real Number

**Example 1:**

Express the given expression in the simplest form without radicals: \(\sqrt[n]{ a <sup>m</sup>}\)

**Solution: **

Given that \(\sqrt[n]{ a<sup>m</sup>}\)

= a<sup>m</sup><sup>\(\frac { 1 }{ n } \)</sup>

= a<sup>m x \(\frac { 1 }{ n } \)</sup>

= a \(\frac { m }{ n } \)

**Example 2: **

Simplify the nth root of the given number: \(\sqrt[3]{ 27 }\)

**Solution:
**Given that, \(\sqrt[3]{ 27 }\)

Let’s find the nth root of it ie.,

\(\sqrt[3]{ 27 }\) = 3

Because cube root of 3 is 3 ie., 3³ = 27

Therefore, the value of \(\sqrt[3]{ 27 }\) is 3.

**Example 3:
**Simplify 343

^{\(\frac { 2 }{ 3 } \)}and write as a radical.

**Solution:
**

Given that 343^{2/3}

As per the rational exponents a^{m/n} = (\(\sqrt[n]{ a }\))^{m} = \(\sqrt[n]{ a^{m} }\))

So, 343^{2/3} = (\(\sqrt[3]{ 343}\))<sup>2</sup> = \(\sqrt[3]{ 343^{2} }\))

We know that \(\sqrt[3]{ 343}\) = 7 because cube root of 7 is 343

\(\sqrt[3]{ 343^{2} }\)) = 7^{2} =49.

### FAQs on How to find the nth root of A without a calculator

**1. What is the nth root called?**

The nth root is called a radical expression or a radical.

**2. How to Solve the nth root of a number?**

**3. What is the expression of radical A?**

The nth root of a is expressed as \(\sqrt[n]{ a }\).

**4. How to find the indicated real nth roots of A?**