nth Roots

Last Updated : 19 Jan, 2021

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?

$\sqrt[n]{}$ indicates an nth root.

For Example:

$\sqrt[n]{144}$
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 $\sqrt{36}$ 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
=> $a=\sqrt[5]{32}$
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
=> $a=\sqrt[6]{4096}$
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 $c= ( \sqrt[n]{a}\ * \sqrt[n]{b})$ then find the value of c?

Solution:

Given: $c = ( \sqrt[n]{a}\ * \sqrt[n]{b})$
SInce the index value is same then the radicand is multiplied
$\\ c=( \sqrt[2]{18} * \sqrt[2]{2})\\ c= \sqrt[2]{36}$
Since, 6 is the square root of 36 (6² = 36)
Therefore, c = 6

Example 4: If a = 5, b = 15, n = 3 and $c = ( \frac{\sqrt[n]{a}}{\sqrt[n]{b}})$ then find the value of c?

Solution:

Given: $c = ( \frac{\sqrt[n]{a}}{\sqrt[n]{b}})$
SInce the index value is same then the radicand are divided
$\\ c = \sqrt[n]{\frac{a}{b}} \\ c = \sqrt[3]{\frac{5}{15}}\\ c = \sqrt[3]{\frac{1}{3}}\\ c = \frac{1}{\sqrt[3]{3}}$

Example 5: Solve the following: $c = \sqrt[3]{1} +\sqrt[3]{ 8}+\sqrt[3]{27}+\sqrt[3]{64}$

Solution:

Given: $c = \sqrt[3]{1} +\sqrt[3]{ 8}+\sqrt[3]{27}+\sqrt[3]{64}\\ c= \sqrt[3]{1^3} +\sqrt[3]{2^3}+\sqrt[3]{3^3}+\sqrt[3]{4^3}$
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