nth Roots

Real numbers are the numbers that include both rational and irrational numbers. Rational numbers such as integers (-2, 0, 1), fractions(1/2, 2.5), and irrational numbers such as √3, π(22/7), etc., are all real numbers. The definition of the nth root of a real number can be stated as:

For any two real numbers a and b, and any positive integer n, if aⁿ = b, then a is an nth root of b. 

For Example:

3⁴ = 81

Where,

a = 3, b = 81, and n = 4

That means that 3 is the fourth root of 81.

How to write nth Roots?

 indicates an nth root. 

For Example:

Where,

144 = Radicand,

√ = Radical Sign, and

n = Index

That means that we are taking the nth root of 144. One can take any value of n, like n = 2, 3, 5, etc. We will be taking that root of 100.

Principal Root

Some numbers have more than one real nth root. For example, 36 has two roots, one is +6 and another is -6. In this case, the non-negative root is called the principal root. 

When no index is given the radical sign indicates the principal root. For example, if we take then we are talking about its principal root which is 6. 

Examples

Example 1: If n = 5 and b = 32  then a = ?

Solution:

Given: a⁵ = 32

=> 

2 . 2 . 2 . 2 . 2 = 32 

Since, 2 is the fifth root of 32

Therefore, a = 2

Example 2: n = 6 and b = 4096 then a = ?

Solution:

Given: a⁶ = 4096

=> 

To find the value of a, we will check which integer value is the sixth root of 4096

2 . 2 . 2 . 2 . 2 . 2 = 64 

3 . 3 . 3 . 3 . 3 . 3 = 729 

4 . 4 . 4 . 4 . 4 . 4 = 4096 

Since, 4 is the sixth root of 4096 

Therefore, a = 4

Example 3: If a = 18, b = 2, n  = 2, and then find the value of c?

Solution:

Given: 

SInce the index value is same then the radicand is multiplied

Since, 6 is the square root of 36 (6² = 36) 

Therefore, c = 6

Example 4: If a = 5, b = 15, n = 3 andthen find the value of c?

Solution:

Given: 

SInce the index value is same then the radicand are divided

Example 5: Solve the following:

Solution:

Given: 

1 . 1 . 1 = 1, 1 is the cube root of 1

2 . 2 . 2 = 8, 2 is the cube root of 8

3 . 3 . 3 = 27, 3 is the cube root of 27

4 . 4 . 4 = 64, 4 is the cube root of 64

c = 1 + 2 + 3 + 4 

c = 10